Physics Book - User contributions [en]

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2019-08-02T15:21:45Z

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__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

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==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transfer of Thermal Energy by Conduction]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Main Page

2019-08-02T15:19:59Z

Seisner6: /* Angular Momentum */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transfer of Thermal Energy by Conduction]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Main Page

2019-08-02T15:16:21Z

Seisner6: /* Angular Momentum */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transfer of Thermal Energy by Conduction]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

The Moments of Inertia

2019-06-25T20:57:53Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore

:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>

:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>

:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''''Solution'''''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

'''Difficult'''

Derive the equation for the rotational inertia of a sphere of mass <math> m </math> and radius <math> r </math> about one of its internal axes. Do this by using the definition of rotational inertia for continuous masses. (Also solve the Riemann Hypothesis)

'''''Solution'''''

In order to solve this problem, we can start by splitting the sphere into smaller objects of known rotational inertia. In this case, the sphere is made up of infinitely many infinitely small disks each centered on the axis of rotation. We know that:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

We then need to sum together the rotational inertia contributions for each of these spheres. Treating, the mass of each infintesimal disk as some <math> dm </math> we have:

<math> I_{sphere} = \frac{1}{2}\int_{-r}^{r} R_{disk}^2 dm </math>

Now, we can express <math> dm </math> in terms of the mass and radius of the sphere by using the mass density of the sphere:

<math> dm = \rho_{sphere}V_{disk} = \rho_{sphere}\pi R_{disk}^2 dz </math>

Using trigonometry, we obtain:

<math> R_{disk}^2 = r^2 - z^2 </math>

So therefore:

<math> dm = \rho_{sphere}\pi(r^2 - z^2) dz </math>

We also know that since the sphere has uniform mass density, we have:

<math> \rho_{sphere} = \frac{m}{\frac{4}{3} \pi r^3} </math>

<math> dm = \frac{m}{\frac{4}{3} \pi r^3}\pi(r^2 - z^2) dz </math>

Now, substituting back into the integral and cancelling terms, we arrive at:

<math> I_{sphere} = \frac{1}{2}\frac{m}{\frac{4}{3} \pi r^3}\pi \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

Evaluating the integral, we get:

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} (\frac{16r^5}{15}) </math>

and finally, we arrive at:

<math> I_{sphere} = \frac{2}{5}mr^2 </math>

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:56:40Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore

:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>

:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>

:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''''Solution'''''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

'''Difficult'''

Derive the equation for the rotational inertia of a sphere of mass <math> m </math> and radius <math> r </math> about one of its internal axes. Do this by using the definition of rotational inertia for continuous masses. (Also solve the Riemann Hypothesis)

'''''Solution'''''

In order to solve this problem, we can start by splitting the sphere into smaller objects of known rotational inertia. In this case, the sphere is made up of infinitely many infinitely small disks each centered on the axis of rotation. We know that:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

We then need to sum together the rotational inertia contributions for each of these spheres. Treating, the mass of each infintesimal disk as some <math> dm </math> we have:

<math> I_{sphere} = \frac{1}{2}\int_{-r}^{r} R_{disk}^2 dm </math>

Now, we can express <math> dm </math> in terms of the mass and radius of the sphere by using the mass density of the sphere:

<math> dm = \rho_{sphere}V_{disk} = \rho_{sphere}\pi R_{disk}^2 dz </math>

Using trigonometry, we obtain:

<math> R_{disk}^2 = r^2 - z^2 </math>

So therefore:

<math> dm = \rho_{sphere}\pi(r^2 - z^2) dz </math>

We also know that since the sphere has uniform mass density, we have:

<math> \rho_{sphere} = \frac{m}{\frac{4}{3} \pi r^3} </math>

<math> dm = \frac{m}{\frac{4}{3} \pi r^3}\pi(r^2 - z^2) dz </math>

Now, substituting back into the integral and cancelling terms, we arrive at:

<math> I_{sphere} = \frac[1}{2}\frac{m}{\frac{4}{3} \pi r^3}\pi \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

Evaluating the integral, we get:

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} (\frac{16r^5}{15} </math>

and finally, we arrive at:

<math> I_{sphere} = \frac{2}{5}mr^2 </math>

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:54:33Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore

:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>

:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>

:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''''Solution'''''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

'''Difficult'''

Derive the equation for the rotational inertia of a sphere of mass <math> m </math> and radius <math> r </math> about one of its internal axes. Do this by using the definition of rotational inertia for continuous masses. (Also solve the Riemann Hypothesis)

'''''Solution'''''

In order to solve this problem, we can start by splitting the sphere into smaller objects of known rotational inertia. In this case, the sphere is made up of infinitely many infinitely small disks each centered on the axis of rotation. We know that:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

We then need to sum together the rotational inertia contributions for each of these spheres. Treating, the mass of each infintesimal disk as some <math> dm </math> we have:

<math> I_{sphere} = \int_{-r}^{r} \frac{1}{2}R_{disk}^2 dm </math>

Now, we can express <math> dm </math> in terms of the mass and radius of the sphere by using the mass density of the sphere:

<math> dm = \rho_{sphere}V_{disk} = \rho_{sphere}\pi R_{disk}^2 dz </math>

Using trigonometry, we obtain:

<math> R_{disk}^2 = r^2 - z^2 </math>

So therefore:

<math> dm = \rho_{sphere}\pi(r^2 - z^2) dz

We also know that since the sphere has uniform mass density, we have:

<math> \rho_{sphere} = \frac{m}{\frac{4}{3} \pi r^3} </math>

<math> dm = \frac{m}{\frac{4}{3} \pi r^3}\pi(r^2 - z^2) dz </math>

Now, substituting back into the integral and cancelling terms, we arrive at:

<math> I_{sphere} = \frac[1}{2}\frac{m}{\frac{4}{3} \pi r^3}\pi \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} \int_{-r}^{r} (r^2 - z^2)^2 dz </math>

Evaluating the integral, we get:

<math> I_{sphere} = \frac{m}{\frac{8}{3} r^3} (\frac{16r^5}{15} </math>

and finally, we arrive at:

<math> I_{sphere} = \frac{2}{5}mr^2 </math>

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:20:44Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''Solution'''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

'''Difficult'''

Solve the Riemann Hypothesis

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:19:46Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''Solution'''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:16:32Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''

What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''

Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''Solution'''

In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:15:52Z

Seisner6: /* Examples */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''
What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''
Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''Solution'''
In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2 </math>

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T20:14:51Z

Seisner6:

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''Simple'''
What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

'''''Solution'''''

For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
:<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
:<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
:<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''Medium'''
Imagine a 0.002 kg ladybug resting on the edge of a spinning disk. If the mass of the disk is 2 kg and it has a radius of 0.15m, what is the combined rotational inertia of the disk and the ladybug about the center of the disk (hint: treat the ladybug as a point mass)?

'''Solution'''
In order to solve this problem, we can write out the combined rotational inertia in terms of it's constituent parts:

<math>I_{tot} = I_{L} + I_{disk} </math>

In this case, we can treat the ladybug as a point mass resting at the edge of the disk, so we have:

<math> I_{L} = M_{L}R_{disk}^2 </math>

<math> I_L = 0.002(0.15)^2 </math>

<math> I_L = 4.5 x 10^{-5} kg \cdot m^2 </math>

Next we calculate the rotational inertia of the disk. As this is an often used object, we know that the rotational inertia of a disk is given by:

<math> I_{disk} = \frac{1}{2}M_{disk}R_{disk}^2 </math>

<math> I_{disk} = \frac{1}{2}(2)(0.15)^2 </math>

<math> I_{disk} = 2.25 x 10^{-2} kg \cdot m^2 </math>

Finally, we have:

<math> I_{tot} = 2.2545 x 10^{-2} kg \cdot m^2

So, we see that in this case the added rotational inertia of the ladybug is negligible compared to the rotational inertia of the disk.

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T00:20:36Z

Seisner6: /* Main Idea */

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{\omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''1. The Moment of Inertia of a Diatomic Molecule'''
What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

:::'''''Solution'''''
:::::::For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
::::::::<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
::::::::<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
::::::::<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''2. '''Calculate the moment of inertia of the array of point masses shown in figure 8-43 below. Assume m=1.8 kg and M=3.1 kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular with the dimensions indicated. The horizontal axis splits the 0.50 m distance in half. The vertical axis is 0.50 m from the left side of the rectangle. [http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf]
[[File:Figure.png|none]]
:::'''''Solution'''''
:::::::''Part A: What is the moment of inertia about the vertical axis?'' <math> I = 1.8kg \times (0.50m)^2 + 1.8kg \times (1.50m)^2 + 3.1kg \times (0.50m)^2 + 3.1kg \times (1.50m)^2 = 12.25 kg\cdot m^2 </math>
:::::::''Part B: What is the moment of inertia about the horizontal axis?'' <math> I = 1.8kg \times (0.25m)^2 + 1.8kg \times (1.50m)^2 + 3.1kg \times (0.25m)^2 + 3.1kg \times (0.25m)^2 = 0.61 kg\cdot m^2 </math>
:::::::''Part C: About which axis would it be harder to accelerate this array?'' The one that has the largest moment of inertia is the hardest to accelerate and that is the vertical axis which has moment of inertia of 12.25 kg <math>\cdot m^2 </math>. Newton's 2nd Law in angular form is <math>\tau = I\alpha </math> so for a given torque <math>\tau</math>, the object with the larger moment of inertia I has the smaller acceleration <math>\alpha</math> since <math>\alpha = \tau/I</math>

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

The Moments of Inertia

2019-06-25T00:19:39Z

Seisner6:

claimed and written by san47
[http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif][[File:Rlin.gif|thumb|300px|Rotation-Linear Parallels]]

==Main Idea==
[[File:Moment_of_inertia_examples.gif|thumb|left|200px|Rotating objects about a chosen axis.]]
Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter ''I''. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from the physical characteristics of the object, specifically it's mass distribution about the axis of rotation. [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html][http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion-]

===A Mathematical Model===
[[File:Point_mass.gif|thumb|right|300px|The point mass model of the moment of inertia.]]
The moment of inertia for a point particle is characterized by the mass of the particle multiplied by the perpindicular radius to the axis of rotation squared, as shown below:

:<math> I = mr^2 </math>

Using this equivalence, it is actually possible to show that the two formulations of angular momentum are equivalent for point particles as well as for continuous masses. For point particles, the proof is quite straightforward:

We start by knowing that:

<math> \vec{L} = \vec{r}\times\vec{p} </math>

and

<math> \vec{L} = I\vec{\omega} </math>

We can now substitute in our expression for I:

<math> \vec{L} = mr^2\vec{\omega} </math>

We can also simplify our first definition of angular momentum by evaluating the cross product. Since the particle is rotating,<math> \vec{r} </math> is by definition perpendicular to <math> \vec{p} </math>, so we arrive at:

<math> \vec{L} = \vec{r}\times\vec{p} = rmv\hat{z} </math>

where <math> \hat{z} </math> is simply the direction perpendicular to both <math> \vec{r} </math> and <math> \vec{p} </math>.

Finally, by noting that <math> \omega = \frac{v}{r} </math>, we can show:

<math> \vec{L} = mr^2\vec{omega} = mr^2\frac{v}{r} = mvr\hat{z} </math>

In doing this, we have shown that the "spin" formulation of angular momentum often used for continuous masses is simply a reformulation of the translational formulation which is used to describe point particles.

'''Extended Masses'''

In the same way that angular momentum could be extended to continuous masses and multiparticle systems, so can the rotational inertia of an object. In fact, it is this extension that gives us the "spin" formulation of angular momentum in the first place

For any extended mass, the rotational inertia can be calculated by taking the limit of the summation used for multiparticle systems as each <math> m_j </math> approaches 0, filling some finite volume with infinitely many of these <math> m_j </math> terms:

<math> I = \sum_{j=1}^n m_jr_j^2 </math>

<math> I = \lim_{m_j \to 0} \sum_{j=1}^\infty m_jr_j^2 </math>

<math> I = \int_M r^2 dm </math>

which can be rewritten in terms of density and volume as:

<math> I = \iiint_V \rho(x, y, z) r^2 dV </math>

==Calculating Moment of Inertia==
[[File:Moment_of_Inertia.jpg|thumb|left|300px|Some common uniform-density solids whose moments of inertia are known.]]
===Thin Rod===
'''''Divide into Small Slices''''' Divide the rod into N small slices of equal length <math>\Delta x = L/N</math>, each with mass of <math>\Delta M = M/N</math>.

'''''The Mass of One Slice''''' Concentrate on one representative slice: <math>N = L/\Delta x</math> so that <math>\Delta M = M/N = M(\Delta x/L)</math>.

'''''The Contribution of One Slice''''' Approximation <math>r_\perp \approx x_n</math>: <math>\Delta I=(\Delta M)x^2 _n = (M/L)x^2 _n\Delta x.</math>

'''''Adding Up the Contributions''''' <math>I = \sum_{n=1}^N \Delta I = (M/L)\sum_{n=1}^N x^2 _n\Delta x </math>.

'''''The Finite Sum Becomes a Definite Integral''''' <math>I = (M/L) \lim_{N \to \infty}\sum_{n=1}^N x^2 _n\Delta x</math> <math>= (M/L) \int\limits_{x_i}^{x_f}x^2\, dx</math>.

'''''The Limits of Integration''''' Since the origin was at the center of the rod: <math>I = (M/L) \int\limits_{-L/2}^{+L/2}x^2\, dx = (1/12)ML^2</math>.

===Hoop===
The moment of inertia of a hoop or thin hollow cylinder of negligible thickness about its central axis is a straightforward extension of the moment of inertia of a point mass since all of the mass is at the same distance R from the central axis.[http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html#ihoop]

===Sphere===
The moment of inertia of a sphere can be expressed as the summation of moments of infinitesimally thin disks about the z axis.
[[File:Sph2.gif|300px|none]] [http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph4]

===Cylinder===
[[File:Cylinder_disk.png|thumb|These disks and cylinders all have moment of inertia <math>1/2MR^2</math>about their axes.]]
The moment of inertia of a solid cylinder can be calculated from the built up of moment of inertia of thin cylindrical shells. [http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html#icyl]
Because only the perpendicular distances of atoms from the axis matter(<math>r_\perp</math>), the moment of inertia for rotation about the axis of a long cylinder has exactly the same form as that of a disk. It means that the moment of inertia does not matter how long the cylinder is or how thick the disk is.

===Other===
The moments of inertia for different shapes can be calculated by applying integral calculus as shown in the calculation of moment of inertia for the [[#section|Thin Rod]] or [[:File:sph2.gif|sphere]].

==Examples==
'''1. The Moment of Inertia of a Diatomic Molecule'''
What is the moment of inertia of a diatomic nitrogen molecule <math>N_2</math> around its center of mass? The mass of a nitrogen atom is <math>2.3 \times 10^{-26} kg</math> and the average distance between nuclei is <math>1.5 \times 10^{-10} m.</math> Use the definition of moment of inertia carefully.

:::'''''Solution'''''
:::::::For two masses, <math> I = m_1r^2 {_\perp,_1} + m_2r^2 {_\perp,_2}</math>. The distance between masses is d, so the distance of each object from the center of mass is <math>r{_\perp,1} = r{_\perp,2} = (d/2)</math>. Therefore
::::::::<math>I = M(d/2)^2 + M(d/2)^2 = 2M(d/2)^2</math>
::::::::<math>I = 2 \cdot (2.3 \times 10^{-26} kg)(0.75 \times 10^{-10} m)^2 </math>
::::::::<math>I = 2.6 \times 10^{-46} kg \cdot m^2</math>

'''2. '''Calculate the moment of inertia of the array of point masses shown in figure 8-43 below. Assume m=1.8 kg and M=3.1 kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular with the dimensions indicated. The horizontal axis splits the 0.50 m distance in half. The vertical axis is 0.50 m from the left side of the rectangle. [http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf]
[[File:Figure.png|none]]
:::'''''Solution'''''
:::::::''Part A: What is the moment of inertia about the vertical axis?'' <math> I = 1.8kg \times (0.50m)^2 + 1.8kg \times (1.50m)^2 + 3.1kg \times (0.50m)^2 + 3.1kg \times (1.50m)^2 = 12.25 kg\cdot m^2 </math>
:::::::''Part B: What is the moment of inertia about the horizontal axis?'' <math> I = 1.8kg \times (0.25m)^2 + 1.8kg \times (1.50m)^2 + 3.1kg \times (0.25m)^2 + 3.1kg \times (0.25m)^2 = 0.61 kg\cdot m^2 </math>
:::::::''Part C: About which axis would it be harder to accelerate this array?'' The one that has the largest moment of inertia is the hardest to accelerate and that is the vertical axis which has moment of inertia of 12.25 kg <math>\cdot m^2 </math>. Newton's 2nd Law in angular form is <math>\tau = I\alpha </math> so for a given torque <math>\tau</math>, the object with the larger moment of inertia I has the smaller acceleration <math>\alpha</math> since <math>\alpha = \tau/I</math>

== See also ==

===External links===
http://www.bsharp.org/physics/spins

http://www.real-world-physics-problems.com/physics-of-figure-skating.html

==References==
# Nave, R. [http://hyperphysics.phy-astr.gsu.edu/hbase/images/inecon.gif "Concepts"] HyperPhysics. Web.
# Nave, R. "Moment of Inertia" [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html HyperPhysics.] Web.
# Urone, Paul Peter., Roger Hinrichs, Kim Dirks, and Manjula Sharma. [http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a@9.4:70/Dynamics-of-Rotational-Motion- "Rotational Inertia and Moment of Inertia."] College Physics. Houston, TX: OpenStax College, Rice U, 2013. 354. Print.
# Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.
#http://www.hunter.cuny.edu/physics/courses/physics110/repository/files/Giancoli_Prob.Solution%20Chap.8.pdf

Total Angular Momentum

2019-06-24T23:30:15Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math>

'''Multiparticle Systems'''

The same technique can also be used to describe the total angular momentum of a multi particle system revolving about a center of rotation. In this case, the total angular momentum is simply the sum of the angular momentum for each particle:

:<math> \vec{L_{tot}} = \sum_{j=1}^n \vec{r_j}\times\vec{p_j} </math>

'''Continuous Masses'''

If you imagine letting the summation for the multiparticle system approach infinity, that is if you were to fill some finite volume with an infinite number of infinitesimally small particles, you would create a continuous distribution of mass rotating about the same central axis. This is how we model the angular momentum of extended objects, or objects that cannot be represented by points. When this is done, you arrive at the following equation:

<math> \vec{L} = I\vec{\omega} </math>

Here <math> I </math> is the rotational inertia of the object (For more info see Rotational Inertia) and <math> \omega </math> is the angular frequency.

It is also Important to note that an object could be treated as a point particle about one axis and an extended object about another, i.e. the Earth rotating while revolving around the sun. In this case, it is useful to separate the object's angular momentum into translational and "spin" components (for more info see Translational & Spin Angular Momentum)

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:28:22Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math>

'''Multiparticle Systems'''

The same technique can also be used to describe the total angular momentum of a multi particle system revolving about a center of rotation. In this case, the total angular momentum is simply the sum of the angular momentum for each particle:

:<math> \vec{L_{tot}} = \sum_{j=1}^n \vec{r_j}\times\vec{p_j} </math>

'''Continuous Masses'''

If you imagine letting the summation for the multiparticle system approach infinity, that is if you were to fill some finite volume with an infinite number of infinitesimally small particles, you would create a continuous distribution of mass rotating about the same central axis. This is how we model the angular momentum of extended objects, or objects that cannot be represented by points. When this is done, you arrive at the following equation:

<math> = \vec{L} = I\vec{\omega} </math>

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:25:22Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math>

'''Multiparticle Systems'''

The same technique can also be used to describe the total angular momentum of a multi particle system revolving about a center of rotation. In this case, the total angular momentum is simply the sum of the angular momentum for each particle:

:<math> \vec{L_tot} = \sum_{j=1}^n \vec{r_j}\times\vec{p_j} </math>

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:24:58Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math>

'''Multiparticle Systems'''

The same technique can also be used to describe the total angular momentum of a multi particle system revolving about a center of rotation. In this case, the total angular momentum is simply the sum of the angular momentum for each particle:

:<math> \vec{L_tot} = \sum_{j=1}^n \vec{r_j}\times\vec{p_j}

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:20:13Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math> (1)

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:19:50Z

Seisner6: /* A Mathematical Model */

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} &= \vec{r}\times\vec{p} </math>

===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:18:53Z

Seisner6:

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:<math> \vec{L} = \vec{r}\times\vec{p} </math>
===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Total Angular Momentum

2019-06-24T23:17:39Z

Seisner6:

Completed by Shaefali Padiyar Spring 2018

==The Main Idea==

[[File:Translational and Rotational Angular Momentum.png]]

In the same way it can be convienent to analyze the linear motion of a system via the changes in the linear momentum of the system, it can be useful to use the concept of angular momentum to describe the motion of rotating systems. In general, momentum is a useful concept because it is usually possible to find a system in which the total momentum of the system is conserved - that is, no external force is acting on the system. However, even when momentum isn't conserved, it still provides a useful relationship between the rate of change of an object's motion and the force applied to the object. Angular momentum is useful for this same reason and can be utilized in a similar way.

===A Mathematical Model===

'''Single Particle Systems'''

For a single particle revolving about a center of rotation, the angular momentum of the particle is expressed in terms of the particle's translational momentum about the center, as shown below:

:{math} \vec{L} = \vec{r}\times\vec{p} {/math}
===A Computational Model===

You can visualize this topic of total angular momentum with this code of the total angular momentum of a binary star. You can see the rotational angular momentum along with the translational angular momentum.

[[File:Screen_Shot_2015-12-02_at_1.36.42_AM.png]]

This screenshot shows the code in action but you can check it out with this link: '''https://trinket.io/glowscript/49129c12fd'''

==Examples==

These examples will help to solidify the difference between the different components of total angular momentum.

===Simple===

A ball falls straight down in the xy plane. Its momentum is shown by the red arrow. What is the direction of the ball's total angular momentum about location A?

[[File:Simple final.png]]

When attempting to solve this question, it is important to recognize that there is no rotational angular momentum, because there is no rotation about the center of mass. The only thing that is happening is that the center of mass is translating from some point A. In this case since there is only translational angular momentum, we would simply find the direction of the total angular momentum by using our right hand rule [hint: r points from A to the ball, and the momentum is pointing in the -y direction]

The direction of the ball's total angular momentum is in the -z direction.

===Middling===

This example shows the importance of understanding the difference between rotational and translational angular momentum.

[[File:Easy.png]]

[[File:simple example.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Screen Shot 2015-11-25 at 3.19.38 PM.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Simple example 1.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
'''

Due to the fact that...

[[File:Total Angular Momentum.png]]

We calculated that our Lrot is zero, our total angular momentum is just based on translational angular momentum. So in this case, total angular momentum = translational angular momentum. So total angular momentum is 9.72 kgm^2/s, with the same direction, into the page.

===Difficult===

This problem shows an example of total angular momentum being based off both translational angular momentum and rotational angular momentum.

[[File:Middle example 1.png]]

[[File:Middle example 2.png]]

'''a) Calculate Lrot (both magnitude and direction)
'''

[[File:Middle rotational.png]]

'''b) Calculate Ltrans,B (both magnitude and direction)
'''

[[File:Middle translational.png]]

'''c) Calculate Ltotal,B (both magnitude and direction)
''

The total angular momentum is rotational + translational so it is <0,0,-11.16> kgm^2/s

==Connectedness==

This topic is connected to my major (biomedical engineering)), because it is important to understand the basics before attempting to solve the bigger picture. In terms of angular momentum, it is important to understand the breakdown of translational and rotational before attempting to solve complex problems involving conservation of angular momentum and more. It can be seen in so many examples, from the rotation of a figure skater to a a yoyo.

Total angular momentum is very interesting to witness in the orbit of planets and satellites.
[[File:Application of Total Angular Momentum.png]]

This image shows the translational angular momentum of the Earth (relative to the location of the Sun) and the rotational angular momentum (relative to the center of mass of the Earth).

It is also important to note that for point particle systems that there is no rotational angular momentum, and only translational angular momentum. This is important, because the total angular momentum for a point particle system is just simply translational angular momentum.

Understanding the topic of total angular momentum is extremely important in applying the conservation of angular momentum. Due to the fact that angular momentum is conserved, then it is important to note that if there is net force on some body directed towards a fix point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant. There is a very useful application of constant angular momentum, specifically seen in dealing with the orbits of planets and satellites. This concept is also used for the Bohr model of the atom.

==History==

Newton, in the Principia, hinted at angular momentum in his examples of the First Law of Motion,

''"A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time."
''
However, his geometric proof of the Law of Areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation — his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

''"...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation."
''

== See also ==

===Further reading===

Books and other print media that cover this topic in more depth:

Elementary Theory of Angular Momentum (Dover Books on Physics) by M.E. Rose

Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood

http://chaos.utexas.edu/wp-uploads/2012/03/Angular_Momentum_21.pdf

===Video Content===

Videos on Total Angular Momentum:

https://www.youtube.com/watch?v=8SfRmqSQENU&index=34&list=PL9HgJKLOnKxedh-yIp7FDzUTwZeTeoR-Y

https://www.youtube.com/watch?v=nFSMu3bxXVA

https://www.youtube.com/watch?v=diZDoY07LG4

https://www.youtube.com/watch?v=iWSu6U0Ujs8

===External links===

These are some internet articles that can show more animations and pictures to help understand this concept

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

==References==

https://www.chegg.com/homework-help

http://www.sparknotes.com/physics/rotationalmotion/angularmomentum/section1.rhtml

http://www.phy.duke.edu/~lee/P53/sys.pdf

http://farside.ph.utexas.edu/teaching/301/lectures/node120.html

https://www.physics.purdue.edu/webapps/index.php/course_document/index/phys172/1332/135/9703

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html

http://www.sparknotes.com/testprep/books/sat2/physics/chapter10section6.rhtml

Professor Gumbart [Georgia Institute of Technology] Lecture Notes

Matter and Interactions 4th edition. Full Citation: Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

https://en.wikipedia.org/wiki/Angular_momentum

[[Category:Which Category did you place this in?]]

Rotational Kinematics

2019-06-14T02:35:27Z

Seisner6: /* Difficult */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

Given an angular acceleration modeled by:

:<math>\alpha = K{e}^{-{\beta}t}</math>

Where <math>K</math>, and <math>\beta</math> are both constants.

Find the equations which model the angular velocity and angular displacement as a function of time.

In this problem, as we have an angular acceleration which varies as a function of time, it is important to note that our uniform angular accelerated motion equations are invalid as they require acceleration be constant. Therefore, we must use the basic relationships between <math>\omega</math>, <math>\theta</math>, and <math>\alpha</math>:

:<math>\alpha = \frac{d\omega}{dt}</math>

:<math>\alpha = \frac{{d}^{2}\omega}{{dt}^{2}}</math>

Therefore, beginning with angular velocity, we have:

:<math>K{e}^{-{\beta}t} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>{\int}K{e}^{-{\beta}t}dt = {\int}\frac{d\boldsymbol{\omega}}{dt}dt</math>

:<math>{\int}K{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

:<math>K{\int}{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

This integral can then be solved via u substitution (for more information on solving Integrals, please see "Insert Help page here"). However, here we will simply give the solution to the integral:

:<math>\omega = \frac{-K}{\beta}{e}^{-{\beta}t}</math>

Next, we must solve for the angular displacement function. To do this, we simply integrate the angular velocity function:

:<math>\omega = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\frac{-K}{\beta}{e}^{-{\beta}t} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t}{dt} = \int\frac{d\boldsymbol{\theta}}{dt}dt</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t}{dt} = \boldsymbol{\theta}</math>

Finally, we arrive at:

:<math>\boldsymbol{\theta} = \frac{K}{{\beta}^{2}}{e}^{-{\beta}t}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:34:20Z

Seisner6: /* Difficult */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

Given an angular acceleration modeled by:

:<math>\alpha = K{e}^{-{\beta}t}</math>

Where <math>K</math>, and <math>\beta</math> are both constants.

Find the equations which model the angular velocity and angular displacement as a function of time.

In this problem, as we have an angular acceleration which varies as a function of time, it is important to note that our uniform angular accelerated motion equations are invalid as they require acceleration be constant. Therefore, we must use the basic relationships between <math>\omega</math>, <math>\theta</math>, and <math>\alpha</math>:

:<math>\alpha = \frac{d\omega}{dt}</math>

:<math>\alpha = \frac{{d}^{2}\omega}{{dt}^{2}}</math>

Therefore, beginning with angular velocity, we have:

:<math>K{e}^{-{\beta}t} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>{\int}K{e}^{-{\beta}t}dt = {\int}\frac{d\boldsymbol{\omega}}{dt}dt</math>

:<math>{\int}K{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

:<math>K{\int}{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

This integral can then be solved via u substitution (for more information on solving Integrals, please see "Insert Help page here"). However, here we will simply give the solution to the integral:

:<math>\omega = \frac{-K}{\beta}{e}^{-{\beta}t}</math>

Next, we must solve for the angular displacement function. To do this, we simply integrate the angular velocity function:

:<math>\omega = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\frac{-K}{\beta}{e}^{-{\beta}t = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t\,{dt} = \int\frac{d\boldsymbol{\theta}}{dt}dt</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t\,{dt} = \boldsymbol{\theta}</math>

Finally, we arrive at:

:<math>\boldsymbol{\theta} = \frac{K}{{\beta}^{2}}{e}^{-{\beta}t}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:33:07Z

Seisner6: /* Difficult */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

Given an angular acceleration modeled by:

:<math>\alpha = K{e}^{-{\beta}t}</math>

Where <math>K</math>, and <math>\beta</math> are both constants.

Find the equations which model the angular velocity and angular displacement as a function of time.

In this problem, as we have an angular acceleration which varies as a function of time, it is important to note that our uniform angular accelerated motion equations are invalid as they require acceleration be constant. Therefore, we must use the basic relationships between <math>\omega</math>, <math>\theta</math>, and <math>\alpha</math>:

:<math>\alpha = \frac{d\omega}{dt}</math>

:<math>\alpha = \frac{{d}^{2}\omega}{{dt}^{2}}</math>

Therefore, beginning with angular velocity, we have:

:<math>K{e}^{-{\beta}t} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>{\int}K{e}^{-{\beta}t}dt = {\int}\frac{d\boldsymbol{\omega}}{dt}dt</math>

:<math>{\int}K{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

:<math>K{\int}{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

This integral can then be solved via u substitution (for more information on solving Integrals, please see "Insert Help page here"). However, here we will simply give the solution to the integral:

:<math>\omega = \frac{-K}{\beta}{e}^{-{\beta}t}</math>

Next, we must solve for the angular displacement function. To do this, we simply integrate the angular velocity function:

:<math>\omega = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\frac{-K}{\beta}{e}^{-{\beta}t = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t\, {dt} = \int\frac{d\boldsymbol{\theta}}{dt}dt</math>

:<math>\int\frac{-K}{\beta}{e}^{-{\beta}t\, {dt} = \boldsymbol{\theta}</math>

Finally, we arrive at:

:<math>\boldsymbol{\theta} = \frac{K}{{\beta}^{2}}{e}^{-{\beta}t}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:18:55Z

Seisner6: /* Difficult */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

Given an angular acceleration modeled by:

:<math>\alpha = K{e}^{-{\beta}t}</math>

Find the equations which model the angular velocity and angular displacement as a function of time.

In this problem, as we have an angular acceleration which varies as a function of time, it is important to note that our uniform angular accelerated motion equations are invalid as they require acceleration be constant. Therefore, we must use the basic relationships between <math>\omega</math>, <math>\theta</math>, and <math>\alpha</math>:

:<math>\alpha = \frac{d\omega}{dt}</math>

:<math>\alpha = \frac{{d}^{2}\omega}{{dt}^{2}}</math>

Therefore, beginning with angular velocity, we have:

:<math>K{e}^{-{\beta}t} = \frac{d\omega}{dt}</math>

:<math>{$$\int}K{e}^{-{\beta}t} = {$$\int}\frac{d\omega}{dt}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:17:20Z

Seisner6: /* Difficult */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

Given an angular acceleration modeled by:

:<math>\alpha = K{e}^{-{\Beta}t}

Find the equations which model the angular velocity and angular displacement as a function of time.

In this problem, as we have an angular acceleration which varies as a function of time, it is important to note that our uniform angular accelerated motion equations are invalid as they require acceleration be constant. Therefore, we must use the basic relationships between <math>\omega</math>, <math>\theta</math>, and <math>\alpha</math>:

:<math>\alpha = \frac{d\omega}{dt}</math>

:<math>\alpha = \frac{{d}^{2}\omega}{{dt}^{2}}</math>

Therefore, beginning with angular velocity, we have:

:<math>K{e}^{-{\Beta}t} = \frac{d\omega}{dt}</math>

:<math>{\integral}K{e}^{-{\Beta}t} = {\integral}\frac{d\omega}{dt}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:03:57Z

Seisner6: /* Medium */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\boldsymbol{\alpha}{20}^{2}</math>

:<math>\boldsymbol{\alpha} = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:02:58Z

Seisner6: /* Medium */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: \frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = \frac{1}{2}\alpha{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = \frac{1}{2}\alpha{20}^{2}</math>

:<math>\alpha = \frac{80\pi}{400}\frac{rad}{s}</math>

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-14T02:02:12Z

Seisner6: /* Medium */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

To begin, this problem, we can start by recognizing that we are given:

:<math>{\omega}_{0} = 0\: frac{rad}{s}</math>

:<math>\theta = 40\pi\: rad</math>

:<math>{\Delta}t = 20\: s</math>

We also are given the information that the angular acceleration is constant during this period of time, so we can use the Uniform angular acceleration equations. In this case, we are given <math>{\omega}_{0}</math>, <math>\theta</math>, and <math>{\Delta}t</math>, and we are attempting to solve for <math>\alpha</math>, so it is best to use:

:<math>\theta = frac{1}{2}\alpha{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>{\theta}_{0} = 0</math>

:<math>40\pi = frac{1}{2}\alpha{20}^{2}</math>

:<math>\alpha = frac{80\pi}{400}frac{rad}{s}</math>

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Main Page

2019-06-14T01:46:39Z

Seisner6: /* Rotations */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Help with VPython====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Kinematics]]
*[[Projectile Motion]]
*[[Relativistic Momentum]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Analytical Prediction]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space]]
*[[3 or More Body Interactions]]
*[[Fluid Mechanics]]
*[[An Application of Gravitational Potential]]
*[[Electric Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Momentum with respect to external Forces]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Heat Capacity]]
*[[Calorific Value(Heat of combustion)]]
*[[Specific Heat Capacity]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Models of Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Moment of Inertia for a cylinder]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Quantized energy levels part II]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Insulators====
<div class="mw-collapsible-content">
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Conductors====
<div class="mw-collapsible-content">
*[[Conductivity]]
*[[Charge Transfer]]
*[[Resistivity]]
*[[Polarization of a conductor]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Rotation

2019-06-14T01:46:05Z

Seisner6: Seisner6 moved page Rotation to Rotational Kinematics: Better fits info on page

#REDIRECT [[Rotational Kinematics]]

Rotational Kinematics

2019-06-14T01:46:05Z

Seisner6: Seisner6 moved page Rotation to Rotational Kinematics: Better fits info on page

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T01:07:23Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\: rad</math>

:<math>{\Delta}t = {24}\: hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T01:06:53Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}\, rad</math>

:<math>{\Delta}t = {24}\, hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T01:04:43Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi}</math> ''rad''

:<math>{\Delta}t = {24}</math> ''hr''

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T01:03:15Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = {2\pi} rad</math>

:<math>{\Delta}t = {24} hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = {\frac{2\pi}{24}\frac{rad}{hr}}\times{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T01:02:15Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = 2\pi rad</math>

:<math>{\Delta}t = 24 hr</math>

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{rad}{hr}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = \times{\frac{2\pi}{24}\frac{rad}{hr}}{\frac{1}{3600}\frac{hr}{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{rad}{s}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:59:33Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = 2\pi</math> Radians

:<math>{\Delta}t = 24</math> Hours

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t</math>, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\Delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}x\frac{1}{3600}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\pi}{43200}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:56:21Z

Seisner6: /* Simple */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

:<math>\Delta\boldsymbol{\theta} = 2\pi</math> Radians

:<math>{\Delta}t = 24</math> Hours

Since we are solving for <math>\boldsymbol{\omega}</math> and we have <math>\Delta\boldsymbol{\theta}</math> and <math>{\Delta}t, we can use the discrete time equation for angular velocity:

:<math>\boldsymbol{{\omega}} = \frac{\Delta\boldsymbol{\theta}}{\delta t}</math>

Substituting in our values, we obtain:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>

We can then simply convert to radians per second:

:<math>\boldsymbol{{\omega}} = \frac{2\pi}{24}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}\cross\frac{1}{3600}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:42:31Z

Seisner6: /* History */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:41:07Z

Seisner6:

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that "Magnitudes are in equilibrium at distances reciprocally proportional to their weights" when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:38:07Z

Seisner6:

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== History ==
The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that "Magnitudes are in equilibrium at distances reciprocally proportional to their weights" when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:
"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:34:39Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = {\omega}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + {\omega}_{0}t + {\theta}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:31:47Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\boldsymbol{\omega} = \boldsymbol{{\omega}}_{0} + \boldsymbol{\alpha}t</math>

:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}{t}^{2} + \boldsymbol{{\omega}}_{0}t + \boldsymbol{{\theta}}_{0}</math>

:<math>\boldsymbol{{\omega}}^{2} = \boldsymbol{\omega}</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:25:16Z

Seisner6: /* A Mathematical Model */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

:<math>\omega = {\omega}_{0} + {\alpha}t</math>

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:18:16Z

Seisner6: /* A Mathematical Model */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Objects undergoing rotational motion, just like objects undergoing linear motion, can be analyzed using kinematics, and in principle, by using the relationships above, it is possible to solve for every aspect of an object's rotational motion. However, depending upon the complexity of the motion, this task may prove extremely difficult, so, more often than not special cases are considered, such as "rotation about a fixed axis" or "uniform circular motion". While these special cases may make simplifications about certain aspects of the object's motion, they still make use of the same equations of motion, and thus are still rotational kinematics problems. A special case of particular importance however is the case in which an object in undergoing uniform angular acceleration. In this case, it is possible to derive equations analogous to the Uniformly accelerated motion equations from linear kinematics. These equations are listed below:

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:14:46Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here :<math>\boldsymbol{\theta}</math> represents angular position, :<math>\boldsymbol{\omega}</math> represents angular velocity, and :<math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\Delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\Delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Angular acceleration is equal to alpha:

:<math>\boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}}</math> ,
where <math>{\boldsymbol{a_t}}</math> is the tangential acceleration of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion.

Rotational Kinetic Energy:
An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.

Rotational Kinetic Energy:

::<math>{KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{ω^2}</math>

Relation to Work and Energy Principle:

The energy principle states:
::<math>{E}_{f} = {E}_{i} + W </math>

We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:14:26Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here :<math>\boldsymbol{\theta}</math> represents angular position, :<math>\boldsymbol{\omega}</math> represents angular velocity, and :<math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{{\delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\Delta\boldsymbol{\omega}}{{\delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Angular acceleration is equal to alpha:

:<math>\boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}}</math> ,
where <math>{\boldsymbol{a_t}}</math> is the tangential acceleration of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion.

Rotational Kinetic Energy:
An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.

Rotational Kinetic Energy:

::<math>{KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{ω^2}</math>

Relation to Work and Energy Principle:

The energy principle states:
::<math>{E}_{f} = {E}_{i} + W </math>

We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:13:31Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

Here :<math>\boldsymbol{\theta}</math> represents angular position, :<math>\boldsymbol{\omega}</math> represents angular velocity, and :<math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

:<math>\boldsymbol{\omega} = \frac{\delta\boldsymbol{\theta}}{{\delta}t}</math>

:<math>\boldsymbol{\alpha} = \frac{\delta\boldsymbol{\omega}}{{\delta}t}</math>
[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Angular acceleration is equal to alpha:

:<math>\boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}}</math> ,
where <math>{\boldsymbol{a_t}}</math> is the tangential acceleration of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion.

Rotational Kinetic Energy:
An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.

Rotational Kinetic Energy:

::<math>{KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{ω^2}</math>

Relation to Work and Energy Principle:

The energy principle states:
::<math>{E}_{f} = {E}_{i} + W </math>

We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web

Rotational Kinematics

2019-06-11T00:05:12Z

Seisner6: /* The Main Idea */

Claimed by Aditya Kuntamukkula 2017

==The Main Idea==

Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

===A Mathematical Model===

Similar to linear motion, angular quantities can be described by a series
of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:

:<math>\vec{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>

[[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]

Angular acceleration is equal to alpha:

:<math>\boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}}</math> ,
where <math>{\boldsymbol{a_t}}</math> is the tangential acceleration of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion.

Rotational Kinetic Energy:
An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.

Rotational Kinetic Energy:

::<math>{KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{ω^2}</math>

Relation to Work and Energy Principle:

The energy principle states:
::<math>{E}_{f} = {E}_{i} + W </math>

We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.

==Examples==

===Simple===

A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.

===Medium===

A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.

:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>

:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>

To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.

===Difficult===

A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>

:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}\frac{\frac{\boldsymbol{miles}}{\boldsymbol{hr}}}{\frac{\boldsymbol{rad}}{\boldsymbol{miles}}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}</math>

:<math>\boldsymbol{{w}} = \boldsymbol{{211200}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}}*\frac{\boldsymbol{1}}{\boldsymbol{2\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{rad}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{hr}}{\boldsymbol{min}} = \frac{\boldsymbol{1760}}{\boldsymbol{\pi}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}</math>

==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.

== See also ==

To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.

===Further reading===

Books, Articles or other print media on this topic

===External links===

Some other resources to further understand rotation are the following:

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

==References==

[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.

[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web