Physics Book - User contributions [en]

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2015-12-08T05:12:38Z

Hjamal3: /* Angular Momentum */

__NOTOC__
Welcome to the Georgia Tech Wiki for Intro Physics. This resources 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!

Looking to make a contribution?
#Pick a specific topic from intro physics
#Add that topic, as a link to a new page, under the appropriate category listed below by editing this page.
#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 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 algebra based intro physics textbook [https://openstaxcollege.org/textbooks/college-physics College Physics]
* 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]

== Organizing Categories ==
These are the broad, overarching categories, that we cover in two semester of introductory physics. You can add subcategories or make a new category as needed. A single topic should direct readers to a page in one of these catagories.

<div class="toccolours mw-collapsible mw-collapsed">
===Interactions===
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
**[[Ball and Spring Model of Matter]]
*[[Escape Velocity]]
*[[Fundamental Interactions]]
*[[Determinism]]
*[[System & Surroundings]]
*[[Free Body Diagram]]
*[[Newton's First Law of Motion]]
*[[Newton's Second Law of Motion]]
*[[Newton's Third Law of Motion]]
*[[Gravitational Force]]
*[[Electric Force]]
*[[Conservation of Energy]]
*[[Conservation of Charge]]
*[[Terminal Speed]]
*[[Simple Harmonic Motion]]
*[[Speed and Velocity]]
*[[Derivation of Average Velocity]]
*[[Acceleration]]
*[[Electric Polarization]]
*[[Perpetual Freefall (Orbit)]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
*[[Center of Mass]]
*[[Spring Force]]
*[[Reaction Time]]
*[[Time Dilation]]
*[[Pauli exclusion principle]]
*[[Interactions of Momentum and Energy Principles]]
*[[Magnus Effect]]
</div>
</div>

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

===Modeling with 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">

===Theory===
<div class="mw-collapsible-content">
*[[Einstein's Theory of Special Relativity]]
*[[Einstein's Theory of General Relativity]]
*[[Quantum Theory]]
*[[Maxwell's Electromagnetic Theory]]
*[[Atomic Theory]]
*[[String Theory]]
*[[Elementary Particles and Particle Physics Theory]]
*[[Law of Gravitation]]
*[[Newton's Laws]]
*[[Higgs field]]
*[[Supersymmetry]]
</div>
</div>

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

===Notable Scientists===
<div class="mw-collapsible-content">
*[[Alexei Alexeyevich Abrikosov]]
*[[Christian Doppler]]
*[[Albert Einstein]]
*[[Ernest Rutherford]]
*[[Joseph Henry]]
*[[Michael Faraday]]
*[[J.J. Thomson]]
*[[James Maxwell]]
*[[Robert Hooke]]
*[[Carl Friedrich Gauss]]
*[[Nikola Tesla]]
*[[Andre Marie Ampere]]
*[[Sir Isaac Newton]]
*[[J. Robert Oppenheimer]]
*[[Oliver Heaviside]]
*[[Rosalind Franklin]]
*[[Enrico Fermi]]
*[[Robert J. Van de Graaff]]
*[[Charles de Coulomb]]
*[[Hans Christian Ørsted]]
*[[Philo Farnsworth]]
*[[Niels Bohr]]
*[[Georg Ohm]]
*[[Leo Szilard]]
*[[Galileo Galilei]]
*[[Gustav Kirchhoff]]
*[[Max Planck]]
*[[Heinrich Hertz]]
*[[Edwin Hall]]
*[[James Watt]]
*[[Count Alessandro Volta]]
*[[Josiah Willard Gibbs]]
*[[Richard Phillips Feynman]]
*[[Sir David Brewster]]
*[[Daniel Bernoulli]]
*[[William Thomson]]
*[[Leonhard Euler]]
*[[Robert Fox Bacher]]
*[[Stephen Hawking]]
*[[Amedeo Avogadro]]
*[[Wilhelm Conrad Roentgen]]
*[[Pierre Laplace]]
*[[Thomas Edison]]
*[[Hendrik Lorentz]]
*[[Jean-Baptiste Biot]]
*[[Lise Meitner]]
*[[Lisa Randall]]
*[[Felix Savart]]
*[[Heinrich Lenz]]
*[[Max Born]]
*[[Archimedes]]
*[[Jean Baptiste Biot]]
*[[Carl Sagan]]
*[[Eugene Wigner]]
*[[Marie Curie]]
*[[Pierre Curie]]
*[[Werner Heisenberg]]
*[[Johannes Diderik van der Waals]]
*[[Louis de Broglie]]
*[[Aristotle]]
*[[Émilie du Châtelet]]
*[[Blaise Pascal]]
*[[Siméon Denis Poisson]]
*[[Benjamin Franklin]]
*[[James Chadwick]]
*[[Henry Cavendish]]
*[[Thomas Young]]
*[[James Prescott Joule]]
*[[John Bardeen]]
*[[Leo Baekeland]]
*[[Alhazen]]
*[[Willebrord Snell]]
*[[Fritz Walther Meissner]]
*[[Johannes Kepler]]
*[[Johann Wilhelm Ritter]]
*[[Philipp Lenard]]
*[[Robert A. Millikan]]
*[[Joseph Louis Gay-Lussac]]
*[[Guglielmo Marconi]]
*[[William Lawrence Bragg]]
*[[Robert Goddard]]
*[[Léon Foucault]]
*[[Henri Poincaré]]
*[[Steven Weinberg]]
*[[Arthur Compton]]
*[[Pythagoras of Samos]]
*[[Subrahmanyan Chandrasekhar]]
*[[Wilhelm Eduard Weber]]
*[[Edmond Becquerel]]
*[[Joseph Rotblat]]
*[[Carl David Anderson]]
*[[Hermann von Helmholtz]]
*[[Nicolas Leonard Sadi Carnot]]
*[[Wallace Carothers]]
*[[David J. Wineland]]
*[[Rudolf Clausius]]
*[[Edward L. Norton]]
*[[Shuji Nakamura]]
*[[Pierre Laplace Pt. 2]]
*[[William B. Shockley]]
*[[Osborne Reynolds]]
*[[Christian Huygens]]
*[[Hans Bethe]]
*[[Erwin Schrodinger]]
*[[Wolfgang Pauli]]
*[[Paul Dirac]]
*[[Bill Nye]]
*[[Arnold Sommerfeld]]
*[[Ernest Lawrence]]
*[[James Franck]]
*[[Chen-Ning Yang]]
*[[Albert A. Michelson & Edward W. Morley]]
*[[George Paget Thomson]]
</div>
</div>

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

===Properties of Matter===
<div class="mw-collapsible-content">
*[[Mass]]
*[[Velocity]]
*[[Relative Velocity]]
*[[Density]]
*[[Charge]]
*[[Spin]]
*[[SI Units]]
*[[Heat Capacity]]
*[[Specific Heat]]
*[[Wavelength]]
*[[Electrical Conductivity/Resistivity]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Inertia]]
*[[Non-Newtonian Fluids]]
*[[Ferrofluids]]
*[[Color]]
*[[Temperature]]
*[[Plasma]]
*[[Electron Mobility]]
</div>
</div>

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

===Contact Interactions===
<div class="mw-collapsible-content">
* [[Young's Modulus]]
* [[Friction]]
* [[Static Friction]]
* [[Tension]]
* [[Hooke's Law]]
*[[Centripetal Force and Curving Motion]]
*[[Compression or Normal Force]]
* [[Length and Stiffness of an Interatomic Bond]]
* [[Speed of Sound in Solids]]
* [[Iterative Prediction of Spring-Mass System]]
* [[Geneva Drives: An Interesting Method of Movement]]
</div>
</div>

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

===Momentum===
<div class="mw-collapsible-content">
* [[Vectors]]
* [[Kinematics]]
* [[Conservation of Momentum]]
* [[Predicting Change in multiple dimensions]]
* [[Derivation of the Momentum Principle]]
* [[Momentum Principle]]
* [[Impulse Momentum]]
* [[Curving Motion]]
* [[Projectile Motion]]
* [[Multi-particle Analysis of Momentum]]
* [[Iterative Prediction]]
* [[Analytical Prediction]]
* [[Newton's Laws and Linear Momentum]]
* [[Net Force]]
* [[Center of Mass]]
* [[Momentum at High Speeds]]
* [[Momentum with respect to external Forces]]
</div>
</div>

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

===Angular Momentum===
<div class="mw-collapsible-content">
* [[The Moments of Inertia]]
* [[Moment of Inertia for a cylinder]]
* [[Rotation]]
* [[Torque]]
* [[Systems with Zero Torque]]
[[Systems with Zero Torque*]]
* [[Systems with Nonzero Torque]]
* [[Torque vs Work]]
* [[Angular Impulse]]
* [[Right Hand Rule]]
* [[Angular Velocity]]
* [[Predicting the Position of a Rotating System]]
* [[Translational Angular Momentum]]
* [[The Angular Momentum Principle]]
* [[Angular Momentum of Multiparticle Systems]]
* [[Rotational Angular Momentum]]
* [[Total Angular Momentum]]
* [[Gyroscopes]]
* [[Angular Momentum Compared to Linear Momentum]]
*[[Torque 2]]
* [[3 Fundamental Principles of Mechanics]]
* [[Eulerian Angles]]

</div>
</div>

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

===Energy===
<div class="mw-collapsible-content">
*[[The Photoelectric Effect]]
*[[Photons]]
*[[The Energy Principle]]
*[[Predicting Change]]
*[[Rest Mass Energy]]
*[[Kinetic Energy]]
*[[Potential Energy]]
**[[Potential Energy for a Magnetic Dipole]]
**[[Potential Energy of a Multiparticle System]]
**[[Potential Energy of Macroscopic Springs]]
**[[Graviational Potential Energy]]
*[[Work]]
**[[Work Done By A Nonconstant Force]]
*[[Work and Energy for an Extended System]]
*[[Thermal Energy]]
*[[Conservation of Energy]]
*[[Electric Potential]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Gravitational Potential Energy]]
*[[Point Particle Systems]]
*[[Real Systems]]
*[[Spring Potential Energy]]
**[[Ball and Spring Model]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
*[[Translational, Rotational and Vibrational Energy]]
*[[Franck-Hertz Experiment]]
*[[Power (Mechanical)]]
*[[Transformation of Energy]]

*[[Energy Graphs]]
**[[Energy graphs and the Bohr model]]
*[[Air Resistance]]
*[[Electronic Energy Levels]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Specific Heat Capacity]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Electronic Energy Levels and Photons]]
*[[Energy Density]]
*[[Bohr Model]]
*[[Quantized energy levels]]
**[[Spontaneous Photon Emission]]
*[[Path Independence of Electric Potential]]
*[[Energy in a Circuit]]
*[[The Photovoltaic Effect]]

</div>
</div>

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

===Collisions===
<div class="mw-collapsible-content">
[[File:opener.png]]

*[[Collisions]]
Collisions are events that happen very frequently in our day-to-day world. In the realm of Physics, a collision is defined as any sort of process in which before and after a short time interval there is little interaction, but during that short time interval there are large interactions. When looking at collisions, it is first important to understand two very important principles: the Momentum Principle and the Energy Principle. Both principles serve use when talking of collisions because they provide a way in which to analyze these collisions. Collisions themselves can be categorized into 3 main different types: elastic collisions, inelastic collisions, maximally inelastic collisions. All 3 collisions will get touched on in more detail further on.
[[File:pe.png]]

*[[Elastic Collisions]]
A collision is deemed "elastic" when the internal energy of the objects in the system does not change (in other words, change in internal energy equals 0). Because in an elastic collision no kinetic energy is converted over to internal energy, in any elastic collision Kfinal always equals Kinitial.
[[File:Elco.png]]

*[[Inelastic Collisions]]
A collision is said to be "inelastic" when it is not elastic; therefore, an inelastic collision is an interaction in which some change in internal energy occurs between the colliding objects (in other words, change in internal energy does not equal 0). Examples of such changes that occur between colliding objects include, but are not limited to, things like they get hot, or they vibrate/rotate, or they deform. Because some of the kinetic energy is converted to internal energy during an inelastic collision, Kfinal does not equal Kinitial.
There are a few characteristics that one can search for when identifying inelasticity. These indications include things such as:
*Objects stick together after the collision
*An object is in an excited state after the collision
*An object becomes deformed after the collision
*The objects become hotter after the collision
*There exists more vibration or rotation after the collision
[[File:inve.gif]]

*[[Maximally Inelastic Collision]]
Maximally inelastic collisions, also known as "sticking collisions", are the most extreme kinds of inelastic collisions. Just as its secondary name implies, a maximally inelastic collision is one in which the colliding objects stick together creating maximum dissipation. This does not automatically mean that the colliding objects stop dead because the law of conservation of momentum. In a maximally inelastic collision, the remaining kinetic energy is present only because total momentum can't change and must be conserved.
[[File:inel.gif]]

*[[Head-on Collision of Equal Masses]]
The easiest way to understand this phenomenon is to look at it through an example. In this case, we can analyze it through the common game of billiards. Taking the two, equally massed billiard balls as the system, we can neglect the small frictional force exerted on the balls by the billiard table. The Momentum Principle states that in this head-on collision of billiard balls the total final momentum in the x direction must equal the total initial momentum. However, this alone does not give us the knowledge to know how the momentum will be divided up between the two balls. Considering the law of conservation of energy, we can more accurately depict what will happen. This will also allow for one to identify what kind of collision occurs (elastic, inelastic, or maximally inelastic). It is important to know that head-on collisions of equal masses do not have a definite type of collision associated with it.
[[File:momentum-real-life-applications-2895.jpg]] [[File:8ball.gif]]

*[[Head-on Collision of Unequal Masses]]
Just as with head-on collisions of equal masses, it is easy to understand head-on collisions of unequal masses by viewing it through an example. Let's take for example two balls of unequal masses like a ping-pong ball and a bowling ball. For the purpose of this example (so as to allow for no friction and no other significant external forces), let's imagine these objects collide in outer space inside an orbiting spacecraft. If there were to be a collision between the two, what would one expect to happen? One could expect to see the ping-pong ball collide with the bowling ball and bounce straight back with a very small change of speed. What one might not expect as much is that the bowling ball also moves, just very slowly. Again, this can all be explained through the conservation of momentum and the conservation of energy.
[[File:mi3e.jpg]]

*[[Frame of Reference]]
In the world of Physics, a frame of reference is the perspective from which a system is observed. It can be stationary or sometimes it can even be moving at a constant velocity. In some rare cases, the frame of reference moves at an nonconstant velocity and is deemed "noninertial" meaning the basic laws of physics do not apply. Continuing with the trend of examples, pretend you are at a train station observing trains as they pass by. From your stationary frame of reference, you observe that the passenger on the train is moving at the same velocity as the train. However, from a moving frame of reference, say from the eyes of the train conductor, he would view the train passengers as "anchored" to the train.
[[File:train.png]]

*[[Scattering: Collisions in 2D and 3D]]
Experiments that involve scattering are often used to study the structure and behavior of atoms, nuclei, as well as of other small particles. In an experiment like such, a beam of particles collides with other particles. If it is an atomic or nuclear collision, we are unable to observe the curving trajectories inside the tiny region of interaction. Instead, we can only truly observe the trajectories before and after the collision. This is only possible because the particles are at a farther distance apart and have a very weak mutual interaction; this essentially means that the particles are moving almost in a straight line. A good example which demonstrates scattering is the collision between an alpha particle (the nucleus of a helium atom) and the nucleus of a gold atom. One will understand this phenomenon more in depth after first understanding the Rutherford Experiment which will get touched on later.

*[[Rutherford Experiment and Atomic Collisions]]
In England in 1911, a famous experiment was performed by a group of scientists led by Mr. Ernest Rutherford. This experiment, later known as "The Rutherford Experiment", was a tremendous breakthrough for its time because it led to the discovery of the nucleus inside the atom. Rutherford's experiment involved the scattering of a high-speed alpha particle (now known as a helium nuclei - 2 protons and 2 neutrons) as it was shot at a thin gold foil (consisting of a nuclei with 79 protons and 118 neutrons). In the experiment, Rutherford and his team discovered that the velocity of the alpha particles was not high enough to allow the particles to make actual contact with the gold nucleus. Although they never actually made contact, it is still deemed a collision because there exists a sizable force between the alpha particle and the gold nucleus over a very short period of time. In conclusion, we say the alpha particle is "scattered" by its interaction with the nucleus of a gold atom and experiments like such are called "scattering" experiments.
[[File:ruthef.jpg]]

*[[Coefficient of Restitution]]
The coefficient of restitution is a measure of the elasticity in a collision. It is the ratio of the differences in velocities before and after the collision. The coefficient is evaluated by taking the difference in the velocities of the colliding objects after the collision and dividing by the difference in the velocities of the colliding objects before the collision.

All of the following information was collected from the Matter and Interactions 4th Edition physics textbook. The book is cited as follows...

Chabay, Ruth W., and Bruce A. Sherwood. "Chapter 10: Collisions." Matter & Interactions. Fourth Edition ed. Wiley, 2015. 383-409. Print.

</div>
</div>

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

===Fields===
<div class="mw-collapsible-content">
* [[Electric Field]] of a
** [[Point Charge]]
** [[Electric Dipole]]
** [[Capacitor]]
** [[Charged Rod]]
** [[Charged Ring]]
** [[Charged Disk]]
** [[Charged Spherical Shell]]
** [[integrating the spherical shell]]
** [[Charged Cylinder]]
**[[A Solid Sphere Charged Throughout Its Volume]]
*[[Charge Density]]
*[[Superposition Principle]]
*[[Electric Potential]]
**[[Potential Difference Path Independence]]
**[[Potential Difference in a Uniform Field]]
**[[Potential Difference of point charge in a non-Uniform Field]]
**[[Potential Difference at One Location]]
**[[Sign of Potential Difference]]
**[[Potential Difference in an Insulator]]
**[[Energy Density and Electric Field]]
** [[Systems of Charged Objects]]
*[[Electric Force]]
*[[Polarization]]
**[[Polarization of an Atom]]
**[[Charged Conductor and Charged Insulator]]
**[[Polarization and Drift Speed]]
*[[Charge Motion in Metals]]
*[[Charge Transfer]]
**[[Electrostatic Discharge]]
*[[Magnetic Field]]
**[[Right-Hand Rule]]
**[[Direction of Magnetic Field]]
**[[Magnetic Field of a Long Straight Wire]]
**[[Magnetic Field of a Loop]]
**[[Magnetic Field of a Solenoid]]
**[[Bar Magnet]]
**[[Magnetic Dipole Moment]]
***[[Stern-Gerlach Experiment]]
**[[Magnetic Torque]]
**[[Magnetic Force]]
***[[Applying Magnetic Force to Currents]]
***[[Magnetic Force in a Moving Reference Frame]]
***[[The Hall Effect]]
**[[Earth's Magnetic Field]]
**[[Atomic Structure of Magnets]]
*[[Combining Electric and Magnetic Forces]]
**[[Hall Effect]]
**[[Lorentz Force]]
**[[Biot-Savart Law]]
**[[Biot-Savart Law for Currents]]
**[[Integration Techniques for Magnetic Field]]
**[[Sparks in Air]]
**[[Motional Emf]]
**[[Detecting a Magnetic Field]]
**[[Moving Point Charge]]
**[[Non-Coulomb Electric Field]]
**[[Electric Motors]]
**[[Solenoid Applications]]
</div>
</div>

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

===Simple Circuits===
<div class="mw-collapsible-content">
*[[Components]]
*[[Steady State]]
*[[Non Steady State]]
*[[Charging and Discharging a Capacitor]]
*[[Work and Power In A Circuit]]
*[[Thin and Thick Wires]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Resistivity]]
*[[Power in a circuit]]
*[[Ammeters,Voltmeters,Ohmmeters]]
*[[Current]]
**[[AC]]
*[[Ohm's Law]]
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[RC]]
*[[Parallel Circuits vs. Series Circuits]]
*[[AC vs DC]]
**[[Rectification (Converting AC to DC)]]
*[[Charge in a RC Circuit]]
*[[Current in a RC circuit]]
*[[Circular Loop of Wire]]
*[[Current in a RL Circuit]]
*[[Current in an LC Circuit]]
*[[RL Circuit]]
*[[Feedback]]
*[[Transformers (Circuits)]]
*[[Resistors and Conductivity]]
*[[Semiconductor Devices]]
*[[Insulators]]
*[[Volt]]
*[[Batteries]]
*[[Three Prong Circuits]]
</div>
</div>

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

===Maxwell's Equations===
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
**[[Electric Fields]]
***[[Examples of Flux Through Surfaces and Objects]]
**[[Magnetic Fields]]
**[[Proof of Gauss's Law]]
*[[Ampere's 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]]
**[[The Differential Form of Ampere's Law]]
*[[Faraday's Law]]
**[[Curly Electric Fields]]
**[[Inductance]]
***[[Transformers (Physics)]]
***[[Energy Density]]
**[[Lenz's Law]]
***[[Lenz Effect and the Jumping Ring]]
**[[Lenz's Rule]]
**[[Motional Emf using Faraday's Law]]
*[[Ampere-Maxwell Law]]
*[[Superconductors]]
**[[Meissner effect]]
</div>
</div>

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

===Radiation===
<div class="mw-collapsible-content">
*[[Producing a Radiative Electric Field]]
*[[Sinusoidal Electromagnetic Radiaton]]
*[[Lenses]]
*[[Energy and Momentum Analysis in Radiation]]
**[[Poynting Vector]]
*[[Electromagnetic Propagation]]
**[[Wavelength and Frequency]]
*[[Snell's Law]]
*[[Effects of Radiation on Matter]]
*[[Light Propagation Through a Medium]]
*[[Light Scaterring: Why is the Sky Blue]]
*[[Light Refraction: Bending of light]]
*[[Cherenkov Radiation]]
*[[Rayleigh Effect]]
*[[Image Formation]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Sound===
<div class="mw-collapsible-content">
*[[Doppler Effect]]
*[[Nature, Behavior, and Properties of Sound]]
*[[Speed of Sound]]
*[[Resonance]]
*[[Sound Barrier]]
*[[Sound Propagation in Water]]
*[[Chladni Plates]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Waves===
<div class="mw-collapsible-content">
*[[Bragg's Law]]
*[[Standing waves]]
*[[Gravitational waves]]
*[[Plasma waves]]
*[[Wave-Particle Duality]]
*[[Electromagnetic Spectrum]]
*[[Color Light Wave]]
*[[X-Rays]]
*[[Rayleigh Wave]]
*[[Pendulum Motion]]
*[[Transverse and Longitudinal Waves]]
*[[Planck's Relation]]
*[[interference]]
*[[Polarization of Waves]]
*[[Angular Resolution]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

===Real Life Applications of Electromagnetic Principles===
<div class="mw-collapsible-content">
*[[Scanning Electron Microscopes]]
*[[Maglev Trains]]
*[[Spark Plugs]]
*[[Metal Detectors]]
*[[Speakers]]
*[[Radios]]
*[[Ampullae of Lorenzini]]
*[[Electrocytes]]
*[[Cyclotron]]
*[[Generator]]
*[[Using Capacitors to Measure Fluid Level]]
*[[Cyclotron]]
*[[Railgun]]
*[[Magnetic Resonance Imaging]]
*[[Electric Eels]]
*[[Windshield Wipers]]
*[[Galvanic Cells]]
*[[Electrolytic Cells]]
*[[Magnetoreception]]
*[[Memory Storage Devices]]
*[[Electric Pickups]]
*[[Inductive Sensors for Traffic Lights]]
</div>
</div>

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

===Optics===
<div class="mw-collapsible-content">
*[[Mirrors]]
*[[Refraction]]
*[[Quantum Properties of Light]]
*[[Lasers]]
*[[Lenses]]
*[[Dispersion and Scattering]]
*[[Telescopes]]
*[[Resolving Power]]

</div>
</div>

== 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 page for review of [[Vectors]] and vector operations

Eulerian Angles

2015-12-08T05:06:33Z

Hjamal3: /* Introduction */

Eulerian Angles

2015-12-08T05:06:00Z

Hjamal3: /* Example of a Gyroscope */

Eulerian Angles

2015-12-08T05:04:54Z

Hjamal3:

Eulerian Angles

2015-12-08T04:44:16Z

Hjamal3:

Eulerian Angles

2015-12-08T04:32:13Z

Hjamal3: /* Eulerian Angles */

==Introduction==
When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position. In the image shown, we can find out the position of an object at any place in the ''xy'' plane.

[[File:AngularVelocityPosition2D.PNG|thumb|Simple angular velocity equation in 2D.]]

Unfortunately, you cannot integrate an angular velocity vector with 3-D components to get the orientation of the body in three dimensional space. This requires the concept of '''Eulerian Angles''', which are used in determining a body's orientation in space using angular velocity in the x, y, and z axes.

===Eulerian Angles===

Essentially, to describe the orientation of a body in 3 dimensions we must describe its rotation about three specific axes. In our example shown in the image, we will first rotate our body's reference frame about the '''k''' axis (black to blue), then the '''j''' axes (blue to green), and finally the '''k''' axis again (green to red).

[[File:RotationsEulerianAngles3D.PNG|thumb|Rotationsabouteulerianangles]]

Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

'''Angular Velocity'''
Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ' '''k (green/red axis)'''. However, everything here is defined in different frames and we must consolidate our equations into one coordinate system.

'''Converting Into One Frame'''

==Example of a Gyroscope==

We can use the procedure outlined above to describe the orientation of a gyroscope at any time given the components of its angular velocity. An example of a rotation gyroscope is shown in the image.

[[File:Gyroscope operation.gif|thumb|Gyroscope operation]]

===Simple===
You drop a single coffee filter of mass 1.1 grams from a very tall building, and it takes 46 seconds to reach the ground. In a small fraction of that time the coffee filter reached terminal speed.

1. What is the upward force of air resistance while the coffee filter was falling at terminal speed?
At terminal velocity Fair=mg
1.1/1000 = .0011kg
.0011kg* 9.8 m/s^2 = .01078N

2. If you drop a stack of 6 coffee filters what is the upward force of air resistance at terminal speed?
1.1/1000= .0011kg*6= .0066kg
.0066kg*9.8m/s^2 = .06468kg

===Middling/Difficult===
Johnathan is driving a 1000kg car down a road at 30m/s when his friend cuts his breaks again and there is no friction between the wheels and road. He can only rely on air resistance to slow his car down since the emergency brake can only be pulled at 2 m/s. If this force is equal to -kv where v is velocity through air and k is 2, when can John pull the emergency break?

1. [[File:FORCEDIAGRAM.jpg]]

2. Sum of Forces = ma
-Kv= 1000a
-Kv = 1000 (dv/dt)
-2v= 1000 (dv/dt)
-dt/500 = (dv/v)

3. Integrate
(-t/500)+c = ln(v)

4. Solve for v
e^((-t/500) *c = v
v(0) = c1*e^(0) = c1
v(0) = 30
v = 30 *e^(-t/500)
v(t) = 30*e ^ (-t/500)
v(2) = 30*e ^ (-t/500)
ln(1/15)= -t/500
t - 500ln15

==Connectedness==
#How is this topic connected to something that you are interested in?
This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.

#How is it connected to your major?
This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.

#Is there an interesting industrial application?
Air resistance is part of a much bigger application of physics known as aerodynamics which is essentially the study of how fluids and gases interact with objects in motion. The most common example of aerodynamics is airplanes. Engineers and scientist have to use the principles of aerodynamics (air resistance/drag force included) in order to determine the shape, engines, wings of a plane. It gets much more complicated when forces such a light, weight, thrust and drag come together.

[[File:Dragplane.jpg]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Dr. Wayne Whiteman
DEFINE THE PROPERTIES OF ANGULAR VELOCITY FOR 3D MOTION
Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Eulerian Angles

2015-12-08T04:30:28Z

Hjamal3: /* Eulerian Angles */

==Introduction==
When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position. In the image shown, we can find out the position of an object at any place in the ''xy'' plane.

[[File:AngularVelocityPosition2D.PNG|thumb|Simple angular velocity equation in 2D.]]

Unfortunately, you cannot integrate an angular velocity vector with 3-D components to get the orientation of the body in three dimensional space. This requires the concept of '''Eulerian Angles''', which are used in determining a body's orientation in space using angular velocity in the x, y, and z axes.

===Eulerian Angles===

Essentially, to describe the orientation of a body in 3 dimensions we must describe its rotation about three specific axes. In our example shown in the image, we will first rotate our body's reference frame about the '''k''' axis (black to blue), then the '''j''' axes (blue to green), and finally the '''k''' axis again (green to red).

[[File:RotationsEulerianAngles3D.PNG|thumb|Rotationsabouteulerianangles]]

Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

'''Angular Velocity'''
Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ' '''k (green/red axis)'''.

'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''
Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.

'''WHY AIR RESISTANCE DEPENDS ON CROSS-SECTIONAL AREA'''
It can be seen from practice that the bigger the cross-sectional area of the object the larger the effect that air resistance has on the object. This due to the fact that air resistance is the result of the collision of an objects surface with the air molecules. This means that the bigger the surface-area the more collisions with air molecules the object will experience and the faster it'll reach terminal velocity. For example a person going sky diving will fall much slower with an open parachute (more surface area, air resistance has a bigger impact) than with a closed parachute (less surface area, air resistance has a smaller impact).

'''WHY AIR RESISTANCE DEPENDS ON SPEED'''
Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

==Example of a Gyroscope==

We can use the procedure outlined above to describe the orientation of a gyroscope at any time given the components of its angular velocity. An example of a rotation gyroscope is shown in the image.

[[File:Gyroscope operation.gif|thumb|Gyroscope operation]]

===Simple===
You drop a single coffee filter of mass 1.1 grams from a very tall building, and it takes 46 seconds to reach the ground. In a small fraction of that time the coffee filter reached terminal speed.

1. What is the upward force of air resistance while the coffee filter was falling at terminal speed?
At terminal velocity Fair=mg
1.1/1000 = .0011kg
.0011kg* 9.8 m/s^2 = .01078N

2. If you drop a stack of 6 coffee filters what is the upward force of air resistance at terminal speed?
1.1/1000= .0011kg*6= .0066kg
.0066kg*9.8m/s^2 = .06468kg

===Middling/Difficult===
Johnathan is driving a 1000kg car down a road at 30m/s when his friend cuts his breaks again and there is no friction between the wheels and road. He can only rely on air resistance to slow his car down since the emergency brake can only be pulled at 2 m/s. If this force is equal to -kv where v is velocity through air and k is 2, when can John pull the emergency break?

1. [[File:FORCEDIAGRAM.jpg]]

2. Sum of Forces = ma
-Kv= 1000a
-Kv = 1000 (dv/dt)
-2v= 1000 (dv/dt)
-dt/500 = (dv/v)

3. Integrate
(-t/500)+c = ln(v)

4. Solve for v
e^((-t/500) *c = v
v(0) = c1*e^(0) = c1
v(0) = 30
v = 30 *e^(-t/500)
v(t) = 30*e ^ (-t/500)
v(2) = 30*e ^ (-t/500)
ln(1/15)= -t/500
t - 500ln15

==Connectedness==
#How is this topic connected to something that you are interested in?
This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.

#How is it connected to your major?
This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.

#Is there an interesting industrial application?
Air resistance is part of a much bigger application of physics known as aerodynamics which is essentially the study of how fluids and gases interact with objects in motion. The most common example of aerodynamics is airplanes. Engineers and scientist have to use the principles of aerodynamics (air resistance/drag force included) in order to determine the shape, engines, wings of a plane. It gets much more complicated when forces such a light, weight, thrust and drag come together.

[[File:Dragplane.jpg]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Dr. Wayne Whiteman
DEFINE THE PROPERTIES OF ANGULAR VELOCITY FOR 3D MOTION
Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Eulerian Angles

2015-12-08T04:29:49Z

Hjamal3:

==Introduction==
When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position. In the image shown, we can find out the position of an object at any place in the ''xy'' plane.

[[File:AngularVelocityPosition2D.PNG|thumb|Simple angular velocity equation in 2D.]]

Unfortunately, you cannot integrate an angular velocity vector with 3-D components to get the orientation of the body in three dimensional space. This requires the concept of '''Eulerian Angles''', which are used in determining a body's orientation in space using angular velocity in the x, y, and z axes.

===Eulerian Angles===

Essentially, to describe the orientation of a body in 3 dimensions we must describe its rotation about three specific axes. In our example shown in the image, we will first rotate our body's reference frame about the '''k''' axis (black to blue), then the '''j''' axes (blue to green), and finally the '''k''' axis again (green to red).

[[File:RotationsEulerianAngles3D.PNG|thumb|Rotationsabouteulerianangles]]

Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

'''Angular Velocity'''
Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ '''k (green/red axis)'''.

'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''
Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.

'''WHY AIR RESISTANCE DEPENDS ON CROSS-SECTIONAL AREA'''
It can be seen from practice that the bigger the cross-sectional area of the object the larger the effect that air resistance has on the object. This due to the fact that air resistance is the result of the collision of an objects surface with the air molecules. This means that the bigger the surface-area the more collisions with air molecules the object will experience and the faster it'll reach terminal velocity. For example a person going sky diving will fall much slower with an open parachute (more surface area, air resistance has a bigger impact) than with a closed parachute (less surface area, air resistance has a smaller impact).

'''WHY AIR RESISTANCE DEPENDS ON SPEED'''
Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

==Example of a Gyroscope==

We can use the procedure outlined above to describe the orientation of a gyroscope at any time given the components of its angular velocity. An example of a rotation gyroscope is shown in the image.

[[File:Gyroscope operation.gif|thumb|Gyroscope operation]]

===Simple===
You drop a single coffee filter of mass 1.1 grams from a very tall building, and it takes 46 seconds to reach the ground. In a small fraction of that time the coffee filter reached terminal speed.

1. What is the upward force of air resistance while the coffee filter was falling at terminal speed?
At terminal velocity Fair=mg
1.1/1000 = .0011kg
.0011kg* 9.8 m/s^2 = .01078N

2. If you drop a stack of 6 coffee filters what is the upward force of air resistance at terminal speed?
1.1/1000= .0011kg*6= .0066kg
.0066kg*9.8m/s^2 = .06468kg

===Middling/Difficult===
Johnathan is driving a 1000kg car down a road at 30m/s when his friend cuts his breaks again and there is no friction between the wheels and road. He can only rely on air resistance to slow his car down since the emergency brake can only be pulled at 2 m/s. If this force is equal to -kv where v is velocity through air and k is 2, when can John pull the emergency break?

1. [[File:FORCEDIAGRAM.jpg]]

2. Sum of Forces = ma
-Kv= 1000a
-Kv = 1000 (dv/dt)
-2v= 1000 (dv/dt)
-dt/500 = (dv/v)

3. Integrate
(-t/500)+c = ln(v)

4. Solve for v
e^((-t/500) *c = v
v(0) = c1*e^(0) = c1
v(0) = 30
v = 30 *e^(-t/500)
v(t) = 30*e ^ (-t/500)
v(2) = 30*e ^ (-t/500)
ln(1/15)= -t/500
t - 500ln15

==Connectedness==
#How is this topic connected to something that you are interested in?
This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.

#How is it connected to your major?
This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.

#Is there an interesting industrial application?
Air resistance is part of a much bigger application of physics known as aerodynamics which is essentially the study of how fluids and gases interact with objects in motion. The most common example of aerodynamics is airplanes. Engineers and scientist have to use the principles of aerodynamics (air resistance/drag force included) in order to determine the shape, engines, wings of a plane. It gets much more complicated when forces such a light, weight, thrust and drag come together.

[[File:Dragplane.jpg]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Dr. Wayne Whiteman
DEFINE THE PROPERTIES OF ANGULAR VELOCITY FOR 3D MOTION
Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Eulerian Angles

2015-12-08T04:17:26Z

Hjamal3:

==Introduction==
When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position. In the image shown, we can find out the position of an object at any place in the ''xy'' plane.

[[File:AngularVelocityPosition2D.PNG|thumb|Simple angular velocity equation in 2D.]]

Unfortunately, you cannot integrate an angular velocity vector with 3-D components to get the orientation of the body in three dimensional space. This requires the concept of '''Eulerian Angles''', which are used in determining a body's orientation in space using angular velocity in the x, y, and z axes.

===Eulerian Angles===

Essentially, to describe the orientation of a body in 3 dimensions we must describe its rotation about three specific axes. In our example shown in the image, we will first rotate our body's reference frame about the '''k''' axis (black to blue), then the '''j''' axes (blue to green), and finally the '''k''' axis again (green to red).

[[File:RotationsEulerianAngles3D.PNG|thumb|Rotationsabouteulerianangles]]

Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

[[File:File.jpg]]

p = density of the air

C_D = drag coefficient (typically between .3 and 1.0)

A = cross-sectional area

v = speed of object

'''WHY AIR RESISTANCE DEPENDS ON THE DENSITY OF THE AIR'''
The density affects the air resistance for expected reasons, the denser the air is the larger the air resistance becomes. The denser the air is the more air molecules the object collides with and faster the object reaches terminal velocity. This implies that in areas with high altitudes where air is "thinner" or less dense such as Colorado there is less air resistance.

'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''
Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.

'''WHY AIR RESISTANCE DEPENDS ON CROSS-SECTIONAL AREA'''
It can be seen from practice that the bigger the cross-sectional area of the object the larger the effect that air resistance has on the object. This due to the fact that air resistance is the result of the collision of an objects surface with the air molecules. This means that the bigger the surface-area the more collisions with air molecules the object will experience and the faster it'll reach terminal velocity. For example a person going sky diving will fall much slower with an open parachute (more surface area, air resistance has a bigger impact) than with a closed parachute (less surface area, air resistance has a smaller impact).

'''WHY AIR RESISTANCE DEPENDS ON SPEED'''
Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
You drop a single coffee filter of mass 1.1 grams from a very tall building, and it takes 46 seconds to reach the ground. In a small fraction of that time the coffee filter reached terminal speed.

1. What is the upward force of air resistance while the coffee filter was falling at terminal speed?
At terminal velocity Fair=mg
1.1/1000 = .0011kg
.0011kg* 9.8 m/s^2 = .01078N

2. If you drop a stack of 6 coffee filters what is the upward force of air resistance at terminal speed?
1.1/1000= .0011kg*6= .0066kg
.0066kg*9.8m/s^2 = .06468kg

===Middling/Difficult===
Johnathan is driving a 1000kg car down a road at 30m/s when his friend cuts his breaks again and there is no friction between the wheels and road. He can only rely on air resistance to slow his car down since the emergency brake can only be pulled at 2 m/s. If this force is equal to -kv where v is velocity through air and k is 2, when can John pull the emergency break?

1. [[File:FORCEDIAGRAM.jpg]]

2. Sum of Forces = ma
-Kv= 1000a
-Kv = 1000 (dv/dt)
-2v= 1000 (dv/dt)
-dt/500 = (dv/v)

3. Integrate
(-t/500)+c = ln(v)

4. Solve for v
e^((-t/500) *c = v
v(0) = c1*e^(0) = c1
v(0) = 30
v = 30 *e^(-t/500)
v(t) = 30*e ^ (-t/500)
v(2) = 30*e ^ (-t/500)
ln(1/15)= -t/500
t - 500ln15

==Connectedness==
#How is this topic connected to something that you are interested in?
This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.

#How is it connected to your major?
This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.

#Is there an interesting industrial application?
Air resistance is part of a much bigger application of physics known as aerodynamics which is essentially the study of how fluids and gases interact with objects in motion. The most common example of aerodynamics is airplanes. Engineers and scientist have to use the principles of aerodynamics (air resistance/drag force included) in order to determine the shape, engines, wings of a plane. It gets much more complicated when forces such a light, weight, thrust and drag come together.

[[File:Dragplane.jpg]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Picture of the Airplane
"What Is Drag?" What Is Drag? Ed. Nancy Hall. NASA, n.d. Web. 07 Dec. 2015. <https://www.grc.nasa.gov/www/k-12/airplane/drag1.html>.

HOW DOES THE DRAG COEFFICIENT AFFECT AIR RESISTANCE
Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Eulerian Angles

2015-12-08T03:53:38Z

Hjamal3:

==Introduction==
When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position.

[[File:AngularVelocityPosition2D.PNG|thumb|Simple angular velocity equation in 2D.]]

===A Mathematical Model===
Air resistance is a force that essentially opposes motion and dissipates energy. Much like other opposing forces, air resistance is dependent on both the speed and the size of the surface area of the object. Many things go into what affects the force of air resistance, and it can be defined by the following equation:

[[File:File.jpg]]

p = density of the air

C_D = drag coefficient (typically between .3 and 1.0)

A = cross-sectional area

v = speed of object

'''WHY AIR RESISTANCE DEPENDS ON THE DENSITY OF THE AIR'''
The density affects the air resistance for expected reasons, the denser the air is the larger the air resistance becomes. The denser the air is the more air molecules the object collides with and faster the object reaches terminal velocity. This implies that in areas with high altitudes where air is "thinner" or less dense such as Colorado there is less air resistance.

'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''
Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.

'''WHY AIR RESISTANCE DEPENDS ON CROSS-SECTIONAL AREA'''
It can be seen from practice that the bigger the cross-sectional area of the object the larger the effect that air resistance has on the object. This due to the fact that air resistance is the result of the collision of an objects surface with the air molecules. This means that the bigger the surface-area the more collisions with air molecules the object will experience and the faster it'll reach terminal velocity. For example a person going sky diving will fall much slower with an open parachute (more surface area, air resistance has a bigger impact) than with a closed parachute (less surface area, air resistance has a smaller impact).

'''WHY AIR RESISTANCE DEPENDS ON SPEED'''
Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
You drop a single coffee filter of mass 1.1 grams from a very tall building, and it takes 46 seconds to reach the ground. In a small fraction of that time the coffee filter reached terminal speed.

1. What is the upward force of air resistance while the coffee filter was falling at terminal speed?
At terminal velocity Fair=mg
1.1/1000 = .0011kg
.0011kg* 9.8 m/s^2 = .01078N

2. If you drop a stack of 6 coffee filters what is the upward force of air resistance at terminal speed?
1.1/1000= .0011kg*6= .0066kg
.0066kg*9.8m/s^2 = .06468kg

===Middling/Difficult===
Johnathan is driving a 1000kg car down a road at 30m/s when his friend cuts his breaks again and there is no friction between the wheels and road. He can only rely on air resistance to slow his car down since the emergency brake can only be pulled at 2 m/s. If this force is equal to -kv where v is velocity through air and k is 2, when can John pull the emergency break?

1. [[File:FORCEDIAGRAM.jpg]]

2. Sum of Forces = ma
-Kv= 1000a
-Kv = 1000 (dv/dt)
-2v= 1000 (dv/dt)
-dt/500 = (dv/v)

3. Integrate
(-t/500)+c = ln(v)

4. Solve for v
e^((-t/500) *c = v
v(0) = c1*e^(0) = c1
v(0) = 30
v = 30 *e^(-t/500)
v(t) = 30*e ^ (-t/500)
v(2) = 30*e ^ (-t/500)
ln(1/15)= -t/500
t - 500ln15

==Connectedness==
#How is this topic connected to something that you are interested in?
This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.

#How is it connected to your major?
This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.

#Is there an interesting industrial application?
Air resistance is part of a much bigger application of physics known as aerodynamics which is essentially the study of how fluids and gases interact with objects in motion. The most common example of aerodynamics is airplanes. Engineers and scientist have to use the principles of aerodynamics (air resistance/drag force included) in order to determine the shape, engines, wings of a plane. It gets much more complicated when forces such a light, weight, thrust and drag come together.

[[File:Dragplane.jpg]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

Picture of the Airplane
"What Is Drag?" What Is Drag? Ed. Nancy Hall. NASA, n.d. Web. 07 Dec. 2015. <https://www.grc.nasa.gov/www/k-12/airplane/drag1.html>.

HOW DOES THE DRAG COEFFICIENT AFFECT AIR RESISTANCE
Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Eulerian Angles

2015-12-08T03:45:33Z

Hjamal3: This page teaches how to describe the orientation of a rigid body in 3D.

When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position.