Physics Book - User contributions [en]

Gyroscopes

2019-07-25T17:55:58Z

Bbreer6: /* A Mathematical Model */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

<math>L_{rot,r} = Iω</math>

Where the rotational angular momentum points horizontal to the rotor.

The <math>L_{rot,r}</math> will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

<math>Ω = \frac{V_{cm}}{r}</math>

Where <math>r</math> is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''<math>L_{support}</math> = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

<math>L_{rot}Ω</math>

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

<math>\Omega = \frac{\tau CM}{L_{rot}} = \frac{rMg}{I \omega}</math>

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments. At their core, gyroscopes are responsible for measuring or maintaining rotational motion, and it is obvious to see how this would be related to aviation, industrial applications, manufacturing, and vehicle development.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Systems with Zero Torque

2019-07-25T17:38:02Z

Bbreer6: /* A Mathematical Model */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \Delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<ref>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</ref>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that <math>L_{tot} = L_{trans} + L_{rot}</math>, and assuming that a change in <math>L_{trans}</math> is trivial, it follows that <math>L_{rot, f} = L_{rot, i}</math>. Specifically, <math>\omega_{f} \cdot I_{f} = \omega_{i} \cdot I_{i} </math>. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<ref>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<ref>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. Specifically, he claimed to have found that "a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time." Mathematically, this area, <math>\frac{1}{2} \cdot r \cdot v \cdot sin(\theta)</math> is proportional to angular momentum <math>r \cdot v \cdot sin(\theta)</math>. It was Newton's endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<references>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</references>

Torque

2019-07-25T17:34:51Z

Bbreer6: /* Solution */

'''This was edited by Kirsten Reynolds (Spring 2018) (Minor edit).'''

Torque is the measure of how much a force acting on an object causes that object to rotate, creating a tendency for the object to rotate about an axis, fulcrum, or pivot. Torque is most commonly classified as "twist", rotational force, or angular force to an object and applying it to a system changes the angular momentum of the system. The effectiveness of torque depends on where the force is applied and the position at which the force acts relative to a location.

==History==

The concept of torque first originated with Archimedes focused study on levers. While he did not invent the lever, his research and work on it caused him to create the block-and-tackle pulley systems, allowing people to use the principle of leverage to lift heavy objects. Building off of this, he explained how torque comes into play with objects that are twisting or rotating around a pivot, just as a lever does around the point of rotation. Using the Law of the Lever and geometric reasoning, Archimedes developed the concept of torque.

In 1884, the term "torque" was introduced into English scientific literature by James Thomson, a notable scientist remembered for his work on the improvement of water wheels, water pumps, and turbines. Before officially introducing the name torque, the twisting or torsional motion was referred to "moment of couple" or "angular force".

==Modeling and Understanding==
===A Mathematical Model===

Torque is the cross product between the distance vector, a vector from the point of pivot (A) to the point where the force is applied, and the force vector. The force vector, <math>{\vec{F}}</math>, is defined about a particular location. <div>
<div style="text-align: center;">[[File:Torque_formula.png |150x40px]]</div>

When applying a force to an object at an angle <math>{θ}</math> to the radius, a different equation is required to capture both the force of the twist and the distance from the pivot point to the place where the force is applied. This equation finds the magnitude of torque exerted by a force, <math>{\vec{F}}</math> relative to a location (A).

<div style="text-align: center;">[[File:torquemag_formula.png]]</div>

For a purely perpendicular force with a force application at <math>{θ}=90{°}</math>, <math>sin{θ}=1</math> and the torque is r<sub>A</sub>F. For a force that is parallel to the lever arm at an angle <math>{θ}=0{°}</math>, <math>sin{θ}=0</math> and the torque is zero.

====Angular Acceleration====
Net torque on a system is also equal to the moment of inertia multiplied by the angular acceleration.
<div style="text-align: center;"><math>\sumτ = Iα</math></div>

====Angular Momentum Principle====
The equation for torque is derived from the Angular Momentum Principle, which states that torque is equal to the change in length over time. Another equation used to represent torque is

<div style="text-align: center;"><math>{Δ}{\vec{L}} = {\vec{τ}} \times {Δ}t</math></div>

====Units====
The SI unit of torque is the newton meter <math>{N{·}m}</math> or joule per radian <math>{J/rad}</math>. These units are produced from the dot product of a force and the distance over which it acts.

====Addition and Subtraction====
If more than one torque acts on an object, these values can be combined to calculate the overall net torque. If the torques make the object spin in opposite directions, they should be subtracted from one another. If the individual torques make an object spin in the same direction, the values should be added together.

===Direction of the Force===
[[File:Directionofforce.png |The angle at which the force is applied on the point of rotation changes the effectiveness of the twist.]]<div></div>
When applying a force to a system, the direction of the force greatly affects the torque and alters the effectiveness of twisting. As seen, a force parallel to the handle or object using to twist another is extremely ineffective and does not produce a torque. When the force only contains a perpendicular component, it is effective at twisting an object.

<br>

===Point of Application of the Force===
[[File:pointofapplication.png|The farther away from the nut the force is applied, the more effective the twist is.]] <div></div>
The point and placement of application of the force on an object also affects who effective the torque is. The further away from the point of rotation that a force is applied, the more effective the twist is. In order to make your twisting most effective, add length to provide more leverage.

<br><br>

===Direction of Torque===
[[File:Righthandrulebar.png|300x300px]]<div></div>
Because torque is a vector quantity, it is important to determine the direction in which torque occurs. The direction of torque is perpendicular to the radius from the axis and the force being applied to the system. The right-hand rule along the axis of rotation can be used to determine the direction of torque, where torque is in the direction your thumb is pointing. Torque is in the same direction of the change in angular velocity.

Put the fingers of the right hand in the direction of the distance vector (r) and curl the fingers in the direction of the force vector (F). The direction of the torque vector will be in the direction that the thumb is pointing towards.

If the motion is counterclockwise, and the thumb is pointing "out" from the page, the direction of torque can be noted with a "bullseye" pictogram. If the motion is clockwise, and the thumb is pointing "into" the page, the direction can be noted with an "X" pictogram.
<div></div>

==Examples==
===Simple===
====Problem====
A force of 50 N is applied to a wrench that is 30 cm in length. Calculate the torque if the fore is applied perpendicular to the wrench.

[[File:exampleone.png|thumb| Problem diagram]]

====Solution====
Using Equation 1,
<math>{\vec{τ}} = {\vec{r}} \times {\vec{F}}</math>, you can plug in the values given for distance from point of rotation to where the force is being applied and for force.
<div><math>{\vec{r}} = 30 cm = 0.3 m</math></div>
<div><math>{\vec{F}} = -50 N</math></div>

<div style="text-align: center;"><math>{\vec{τ}} = (-50 N) × (0.3 m)</math></div>
<div style="text-align: center;"><math>{\vec{τ}} = -15 N{·}m</math></div>

===Middling===
====Problem====
[[File:exampletwo.png]]

====Solution====
Using Equation 2, plug in the values for <math>r_A</math>, <math>F</math>, and θ.

<div style="text-align: center;"><math>|τ| = (0.35 m) × (16 N) × sin(61°)</math></div>

<div style="text-align: center;"><math>|{\vec{τ}}| = 4.9 N · m</math></div>

===Difficult===
====Problem====
[[File:examplethree.png|center]]

====Solution====

First, create a free body diagram to include all of the forces acting on the system. Using the standard coordinate system, the pivot location is A.
<div></div>
[[File:examplethreesolution.png|center]]

In order to get the net torque, calculate all of the individual torques about the location A and add them up.
<div></div>
Torque due to child 1:
<div></div>
[[File:torquechildone.png|center]]
<div></div>
The magnitude of the torque of child 1 can be found by putting the distance vector and the force vector tail to tail and applying the right hand rule. It is found that the direction of the torque is into the page in a -z direction.
<div></div>
[[File:magtorquechildone.png|center]]

<div></div>
Torque due to child 2:
<div></div>
[[File:torquechildtwo.png|center]]

<div></div>
The cross product of normal force, <math>{\vec{F}}_n</math>, and total direction vector, <math>{\vec{r}}_n,A</math>, can be used to find the net torque about the point of rotation, A. The torque due to the normal force is zero because the force acts at location A, so it can't twist the seesaw.
<div></div>
[[File:nettorqueA.png|center]]
<div></div>

The net torque of the system is:
<div></div>
[[File:finalnettorque.png|center]]

For a brief introduction on a different method of calculating cross products (using matrices and cofactor expansion), watch this video:
[https://youtu.be/-pFvtQbxA0o]
(Useful for when vectors don't happen to line up neatly on the XYZ planes).

"radius" vector in the video refers to the distance vector (r).

==Connectedness==
Torque exists almost everywhere we go and is involved in nearly everything we do. If torque didn't exist, we would only be able to do things linearly in a uniform line and there wouldn't be spin, turn, or circular motion. Actions such as turning a steering wheel or opening a bottle would be impossible without the twisting motion we call torque.

The concept of torque interests me greatly as a Chemical Engineering major because torque plays a role in most chemical processes and unit operations. Using torque for reactions can alter flow rates, create shaft work, and affect the energy balances of continuous, steady state systems.

Torque has many industrial applications in industries such as aerospace, automotive, material processing, medical, robotics, oil and gas, and assembly. It is often used in finishing off materials with operations such as polishing, grinding, and deburring. Torque sensors are often used to determine the amount of power of engines, motors, turbines, and other rotating devices and the sensors make the required torque measurements automatically on many assembly and sure machines.

== See also ==

===Further reading===

[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html Torque] <div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html Angular Momentum] <div></div>
[https://en.wikipedia.org/wiki/Torque_sensor Torque Sensors] <div></div>

===External links===
[https://www.youtube.com/watch?v=QhuJn8YBtmg Video Tutorial on Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.example.RHR.html Practice Problems]<div></div>
[http://online.unitconverterpro.com/list.php?cat=torque Units of Torque]<div></div>

==References==
Chapter 11 of [https://books.google.com/books?id=Gz4HBgAAQBAJ&pg=PA544&lpg=PA544&dq=matter+and+interactions+4th+edition+torque&source=bl&ots=ShdH7G8bcV&sig=uEQbxhpX3-UqcQf4ilXjp2reG5s&hl=en&sa=X&ved=0ahUKEwin-JTU6MTJAhWDQCYKHUoLA00Q6AEIMjAD#v=onepage&q&f=false Matter & Interactions 4th Edition]<div></div>

[http://www.mikeraugh.org/Talks/UNM-2012-LawOfTheLever.pdf Law of the Lever]<div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html Physics of Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html What is Torque?]<div></div>

Torque

2019-07-25T17:34:28Z

Bbreer6: /* Angular Momentum Principle */

'''This was edited by Kirsten Reynolds (Spring 2018) (Minor edit).'''

Torque is the measure of how much a force acting on an object causes that object to rotate, creating a tendency for the object to rotate about an axis, fulcrum, or pivot. Torque is most commonly classified as "twist", rotational force, or angular force to an object and applying it to a system changes the angular momentum of the system. The effectiveness of torque depends on where the force is applied and the position at which the force acts relative to a location.

==History==

The concept of torque first originated with Archimedes focused study on levers. While he did not invent the lever, his research and work on it caused him to create the block-and-tackle pulley systems, allowing people to use the principle of leverage to lift heavy objects. Building off of this, he explained how torque comes into play with objects that are twisting or rotating around a pivot, just as a lever does around the point of rotation. Using the Law of the Lever and geometric reasoning, Archimedes developed the concept of torque.

In 1884, the term "torque" was introduced into English scientific literature by James Thomson, a notable scientist remembered for his work on the improvement of water wheels, water pumps, and turbines. Before officially introducing the name torque, the twisting or torsional motion was referred to "moment of couple" or "angular force".

==Modeling and Understanding==
===A Mathematical Model===

Torque is the cross product between the distance vector, a vector from the point of pivot (A) to the point where the force is applied, and the force vector. The force vector, <math>{\vec{F}}</math>, is defined about a particular location. <div>
<div style="text-align: center;">[[File:Torque_formula.png |150x40px]]</div>

When applying a force to an object at an angle <math>{θ}</math> to the radius, a different equation is required to capture both the force of the twist and the distance from the pivot point to the place where the force is applied. This equation finds the magnitude of torque exerted by a force, <math>{\vec{F}}</math> relative to a location (A).

<div style="text-align: center;">[[File:torquemag_formula.png]]</div>

For a purely perpendicular force with a force application at <math>{θ}=90{°}</math>, <math>sin{θ}=1</math> and the torque is r<sub>A</sub>F. For a force that is parallel to the lever arm at an angle <math>{θ}=0{°}</math>, <math>sin{θ}=0</math> and the torque is zero.

====Angular Acceleration====
Net torque on a system is also equal to the moment of inertia multiplied by the angular acceleration.
<div style="text-align: center;"><math>\sumτ = Iα</math></div>

====Angular Momentum Principle====
The equation for torque is derived from the Angular Momentum Principle, which states that torque is equal to the change in length over time. Another equation used to represent torque is

<div style="text-align: center;"><math>{Δ}{\vec{L}} = {\vec{τ}} \times {Δ}t</math></div>

====Units====
The SI unit of torque is the newton meter <math>{N{·}m}</math> or joule per radian <math>{J/rad}</math>. These units are produced from the dot product of a force and the distance over which it acts.

====Addition and Subtraction====
If more than one torque acts on an object, these values can be combined to calculate the overall net torque. If the torques make the object spin in opposite directions, they should be subtracted from one another. If the individual torques make an object spin in the same direction, the values should be added together.

===Direction of the Force===
[[File:Directionofforce.png |The angle at which the force is applied on the point of rotation changes the effectiveness of the twist.]]<div></div>
When applying a force to a system, the direction of the force greatly affects the torque and alters the effectiveness of twisting. As seen, a force parallel to the handle or object using to twist another is extremely ineffective and does not produce a torque. When the force only contains a perpendicular component, it is effective at twisting an object.

<br>

===Point of Application of the Force===
[[File:pointofapplication.png|The farther away from the nut the force is applied, the more effective the twist is.]] <div></div>
The point and placement of application of the force on an object also affects who effective the torque is. The further away from the point of rotation that a force is applied, the more effective the twist is. In order to make your twisting most effective, add length to provide more leverage.

<br><br>

===Direction of Torque===
[[File:Righthandrulebar.png|300x300px]]<div></div>
Because torque is a vector quantity, it is important to determine the direction in which torque occurs. The direction of torque is perpendicular to the radius from the axis and the force being applied to the system. The right-hand rule along the axis of rotation can be used to determine the direction of torque, where torque is in the direction your thumb is pointing. Torque is in the same direction of the change in angular velocity.

Put the fingers of the right hand in the direction of the distance vector (r) and curl the fingers in the direction of the force vector (F). The direction of the torque vector will be in the direction that the thumb is pointing towards.

If the motion is counterclockwise, and the thumb is pointing "out" from the page, the direction of torque can be noted with a "bullseye" pictogram. If the motion is clockwise, and the thumb is pointing "into" the page, the direction can be noted with an "X" pictogram.
<div></div>

==Examples==
===Simple===
====Problem====
A force of 50 N is applied to a wrench that is 30 cm in length. Calculate the torque if the fore is applied perpendicular to the wrench.

[[File:exampleone.png|thumb| Problem diagram]]

====Solution====
Using Equation 1,
<math>{\vec{τ}} = {\vec{r}} x {\vec{F}}</math>, you can plug in the values given for distance from point of rotation to where the force is being applied and for force.
<div><math>{\vec{r}} = 30 cm = 0.3 m</math></div>
<div><math>{\vec{F}} = -50 N</math></div>

<div style="text-align: center;"><math>{\vec{τ}} = (-50 N) × (0.3 m)</math></div>
<div style="text-align: center;"><math>{\vec{τ}} = -15 N{·}m</math></div>

===Middling===
====Problem====
[[File:exampletwo.png]]

====Solution====
Using Equation 2, plug in the values for <math>r_A</math>, <math>F</math>, and θ.

<div style="text-align: center;"><math>|τ| = (0.35 m) × (16 N) × sin(61°)</math></div>

<div style="text-align: center;"><math>|{\vec{τ}}| = 4.9 N · m</math></div>

===Difficult===
====Problem====
[[File:examplethree.png|center]]

====Solution====

First, create a free body diagram to include all of the forces acting on the system. Using the standard coordinate system, the pivot location is A.
<div></div>
[[File:examplethreesolution.png|center]]

In order to get the net torque, calculate all of the individual torques about the location A and add them up.
<div></div>
Torque due to child 1:
<div></div>
[[File:torquechildone.png|center]]
<div></div>
The magnitude of the torque of child 1 can be found by putting the distance vector and the force vector tail to tail and applying the right hand rule. It is found that the direction of the torque is into the page in a -z direction.
<div></div>
[[File:magtorquechildone.png|center]]

<div></div>
Torque due to child 2:
<div></div>
[[File:torquechildtwo.png|center]]

<div></div>
The cross product of normal force, <math>{\vec{F}}_n</math>, and total direction vector, <math>{\vec{r}}_n,A</math>, can be used to find the net torque about the point of rotation, A. The torque due to the normal force is zero because the force acts at location A, so it can't twist the seesaw.
<div></div>
[[File:nettorqueA.png|center]]
<div></div>

The net torque of the system is:
<div></div>
[[File:finalnettorque.png|center]]

For a brief introduction on a different method of calculating cross products (using matrices and cofactor expansion), watch this video:
[https://youtu.be/-pFvtQbxA0o]
(Useful for when vectors don't happen to line up neatly on the XYZ planes).

"radius" vector in the video refers to the distance vector (r).

==Connectedness==
Torque exists almost everywhere we go and is involved in nearly everything we do. If torque didn't exist, we would only be able to do things linearly in a uniform line and there wouldn't be spin, turn, or circular motion. Actions such as turning a steering wheel or opening a bottle would be impossible without the twisting motion we call torque.

The concept of torque interests me greatly as a Chemical Engineering major because torque plays a role in most chemical processes and unit operations. Using torque for reactions can alter flow rates, create shaft work, and affect the energy balances of continuous, steady state systems.

Torque has many industrial applications in industries such as aerospace, automotive, material processing, medical, robotics, oil and gas, and assembly. It is often used in finishing off materials with operations such as polishing, grinding, and deburring. Torque sensors are often used to determine the amount of power of engines, motors, turbines, and other rotating devices and the sensors make the required torque measurements automatically on many assembly and sure machines.

== See also ==

===Further reading===

[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html Torque] <div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html Angular Momentum] <div></div>
[https://en.wikipedia.org/wiki/Torque_sensor Torque Sensors] <div></div>

===External links===
[https://www.youtube.com/watch?v=QhuJn8YBtmg Video Tutorial on Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.example.RHR.html Practice Problems]<div></div>
[http://online.unitconverterpro.com/list.php?cat=torque Units of Torque]<div></div>

==References==
Chapter 11 of [https://books.google.com/books?id=Gz4HBgAAQBAJ&pg=PA544&lpg=PA544&dq=matter+and+interactions+4th+edition+torque&source=bl&ots=ShdH7G8bcV&sig=uEQbxhpX3-UqcQf4ilXjp2reG5s&hl=en&sa=X&ved=0ahUKEwin-JTU6MTJAhWDQCYKHUoLA00Q6AEIMjAD#v=onepage&q&f=false Matter & Interactions 4th Edition]<div></div>

[http://www.mikeraugh.org/Talks/UNM-2012-LawOfTheLever.pdf Law of the Lever]<div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html Physics of Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html What is Torque?]<div></div>

Torque vs Work

2019-07-25T17:33:17Z

Bbreer6: /* References */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \Delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} = <2, 0, 0> m \times <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m \times <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F} \cdot \vec{r} </math>

<math> W_{1} = <30, 0, 0> N \cdot <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N \cdot <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N \cdot m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m \times <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N \cdot m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

[[Category: Angular Momentum]]

Torque vs Work

2019-07-25T17:32:57Z

Bbreer6: /* Difficult Torque */

Torque vs Work

2019-07-25T17:32:25Z

Bbreer6: /* Middling Work */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \Delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} = <2, 0, 0> m \times <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m \times <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F} \cdot \vec{r} </math>

<math> W_{1} = <30, 0, 0> N \cdot <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N \cdot <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N \cdot m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Torque vs Work

2019-07-25T17:29:54Z

Bbreer6: /* Simple Torque */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \Delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} = <2, 0, 0> m \times <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m \times <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Torque vs Work

2019-07-25T17:29:33Z

Bbreer6: /* Nuances in Torque */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \Delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} \times \vec{F} = <2, 0, 0> m \times <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Torque vs Work

2019-07-25T17:29:05Z

Bbreer6: /* Mathematical Model: The Concepts */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \Delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} X \vec{F} = <2, 0, 0> m X <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Torque vs Work

2019-07-25T17:28:46Z

Bbreer6: /* Mathematical Model: The Concepts */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r_{cm}} \times \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F} \cdot \vec{r} </math>

<i>where '''W''' is the scalar dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scalar value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = \delta K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} X \vec{F} = <2, 0, 0> m X <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Systems with Zero Torque

2019-07-25T17:21:21Z

Bbreer6: /* Examples */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<ref>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</ref>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that <math>L_{tot} = L_{trans} + L_{rot}</math>, and assuming that a change in <math>L_{trans}</math> is trivial, it follows that <math>L_{rot, f} = L_{rot, i}</math>. Specifically, <math>\omega_{f} \cdot I_{f} = \omega_{i} \cdot I_{i} </math>. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<ref>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<ref>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. Specifically, he claimed to have found that "a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time." Mathematically, this area, <math>\frac{1}{2} \cdot r \cdot v \cdot sin(\theta)</math> is proportional to angular momentum <math>r \cdot v \cdot sin(\theta)</math>. It was Newton's endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<references>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</references>

Systems with Zero Torque

2019-07-25T17:20:19Z

Bbreer6: /* History */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that <math>L_{tot} = L_{trans} + L_{rot}</math>, and assuming that a change in <math>L_{trans}</math> is trivial, it follows that <math>L_{rot, f} = L_{rot, i}</math>. Specifically, <math>\omega_{f} \cdot I_{f} = \omega_{i} \cdot I_{i} </math>. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<ref>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<ref>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. Specifically, he claimed to have found that "a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time." Mathematically, this area, <math>\frac{1}{2} \cdot r \cdot v \cdot sin(\theta)</math> is proportional to angular momentum <math>r \cdot v \cdot sin(\theta)</math>. It was Newton's endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<references>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</references>

Systems with Zero Torque

2019-07-25T17:18:11Z

Bbreer6: /* Connectedness */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that <math>L_{tot} = L_{trans} + L_{rot}</math>, and assuming that a change in <math>L_{trans}</math> is trivial, it follows that <math>L_{rot, f} = L_{rot, i}</math>. Specifically, <math>\omega_{f} \cdot I_{f} = \omega_{i} \cdot I_{i} </math>. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <ref>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<ref>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<references>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</references>

Systems with Zero Torque

2019-07-25T17:15:07Z

Bbreer6: /* References */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<references>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</references>

Systems with Zero Torque

2019-07-25T17:14:51Z

Bbreer6: /* References */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

<ref>

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.
</ref>

Systems with Zero Torque

2019-07-25T17:13:18Z

Bbreer6: /* Computation Model */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

We refer the reader to a section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.

Systems with Zero Torque

2019-07-25T17:13:01Z

Bbreer6: /* Examples */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

see corresponding section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2. An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.

Systems with Zero Torque

2019-07-25T17:12:43Z

Bbreer6: /* A Mathematical Model */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, <math>LA_{f} = LA_{i} + r_{net} \cdot \delta t </math> that for systems with zero torque, <math>LA_{f} = LA_{i}</math>.

Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the forces when adding to net force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with <b>zero torque </b>
[[File:Torque_7.JPG]]

== Computation Model ==

see corresponding section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2.An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.

Torque 2

2019-07-25T17:08:25Z

Bbreer6: /* A Computational Model */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as <math>\tau = r_{A} \times F</math> .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogous to <math>F_{net}</math> in regular momentum calculations. Just how a collection of forces acting on a system is summed and called <math>F_{net}</math>, a collection of torques acting on a system is <math>\tau_{net}</math>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
<math>\tau_{net} = r_{A} \cdot F \cdot sin(\theta)</math>

Torque can also be defined as <math>\tau = \frac{dL}{dt}</math> , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of <math>r_{A}</math> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, <math>t_{net}</math> is the total torque, <math>\delta t</math> is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:07:46Z

Bbreer6: /* A Mathematical Model */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as <math>\tau = r_{A} \times F</math> .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogous to <math>F_{net}</math> in regular momentum calculations. Just how a collection of forces acting on a system is summed and called <math>F_{net}</math>, a collection of torques acting on a system is <math>\tau_{net}</math>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
<math>\tau_{net} = r_{A} \cdot F \cdot sin(\theta)</math>

Torque can also be defined as <math>\tau = \frac{dL}{dt}</math> , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of <math>r_{A}</math> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:06:03Z

Bbreer6: /* The Main Idea */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as <math>\tau = r_{A} \times F</math> .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogous to <math>F_{net}</math> in regular momentum calculations. Just how a collection of forces acting on a system is summed and called <math>F_{net}</math>, a collection of torques acting on a system is <math>\tau_{net}</math>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:05:36Z

Bbreer6: /* The Main Idea */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as <math>\tau = r_{A} \ccross F</math> .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogous to <math>F_{net}</math> in regular momentum calculations. Just how a collection of forces acting on a system is summed and called <math>F_{net}</math>, a collection of torques acting on a system is <math>\tau_{net}</math>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:05:24Z

Bbreer6: /* The Main Idea */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as <math>\tau = r_{A} x F</math> .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogous to <math>F_{net}</math> in regular momentum calculations. Just how a collection of forces acting on a system is summed and called <math>F_{net}</math>, a collection of torques acting on a system is <math>\tau_{net}</math>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:03:04Z

Bbreer6: /* Torque */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>). Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:02:56Z

Bbreer6: /* Torque */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau (<math>\tau</math>. Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:01:40Z

Bbreer6: /* References */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau. Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:01:11Z

Bbreer6: /* Further reading */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau. Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

This section contains the the references you used while writing this page
[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T17:00:01Z

Bbreer6: /* A Computational Model */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau. Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vPython, all you would need to do is update position and angular momentum. The code would look something like this:

While t < some number:

<math>L = L + t_{net}\cdot \delta t</math>

<math>v = \frac{L}{mass}</math>

<math>r = r + v \cdot \delta t</math>

<math>t = t + \delta t</math>

where L is the angular momentum initialized earlier, t_{net} is the total torque, \delta t is the time step, and r is the original position.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

Books, Articles or other print media on this topic

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

This section contains the the references you used while writing this page
[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Torque 2

2019-07-25T05:37:14Z

Bbreer6: /* A Mathematical Model */

== Torque ==

Torque is a Latin word that roughly means "twist," and is usually symbolized by the lower case Greek letter tau. Torque is the measurement of how much a force, F, acting on an object will cause that object to rotate. This force is usually applied to an arm of some sort that is attached to a fulcrum or pivot point. For example, using a wrench to loosen or tighten a nut requires the use of torque - where the wrench would be the arm you apply the force to and the nut would be the pivot point that the force rotates around.

[[File:wrench_gif.gif]]

==The Main Idea==

In "Matter & Interactions, Fourth Edition," torque is defined as τ = r<sub>A</sub> x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to F<sub>net</sub> in regular momentum calculations. Just like how a collection of forces acting on a system is called F<sub>net</sub>, a collection of torques acting on a system is τ<sub>net</sub>.

===A Mathematical Model===

[[File:torque_diagram.gif]]

Torque is defined as the cross product of the distance vector, the distance from pivot point to the location of the applied force, with an applied force. The magnitude of torque can be defined as such:
*τ<sub>A</sub> = r<sub>A</sub>Fsinθ
Torque can also be defined as τ = dL/dt , or the derivative of angular momentum, L.
To determine the direction of torque, one can either compute the cross product or apply the "right-hand rule." To use the "right-hand rule," point your fingers in the direction of r<sub>A</sub> and curl your fingers in the direction of F.

[[File:Rhr_torque.JPG]]

If your thumb points up, the force is coming out of the page and is in the positive z-directon. If your thumb points down, the force is going into the page and is in the negative z-direction. This is a right-handed system, and a good way to remember this is by the right-hand rule. This can also be applied to a "left-handed" system, but it is easier to first apply the right-hand rule and then flip the sign on one of the vector magnitudes to account for the different orientation (non-right-handed orientation).

===A Computational Model===

[[File:Torque.gif]]

The above picture is a great example of what torque would look like in the real world. To make a similar simulation in vpython, all you would need to do is update position and angular momentum. The code would look something like this:

While t<some number:

L=L+tnet*deltat

v=L/mass

r=r+v*deltat

t=t+deltat

Where L is your angular momentum initialized earlier, tnet is total torque, deltat is your time step, and r is your original location.

==Examples==

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

===Simple===

In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?

[[File:torqueE1.gif]]

(50N)(1.0m)sin(90) = 50 Nm
===Middling===

A uniform solid disk with radius 9 cm has mass 0.5 kg (moment of inertia I = ½MR2). A constant force 12 N is applied as shown. At the instant shown, the angular velocity of the disk is 25 radians/s in the −z direction (where +x is to the right, +y is up, and +z is out of the page, toward you). The length of the string d is 14 cm.

[[File:disk torque.jpg]]

What are the magnitude and direction of the torque on the disk, about the center of mass of the disk?

(12N)(0.09m)= 1.08 Nm

Direction: -z direction because of the right hand rule

===Difficult===

We want to place another mass on the see-saw to keep the see-saw from tipping. The only other one we have is a 5.0kg mass. Where would we place this to balance the original 3kg mass that was placed 2.00m to the right of the pivot point?

[[File:prob2.jpg]]

(3kg)(9.8 m/s^2)(2.00m) = (5kg)(9.8 m/s^2)(x)

58.8 = 49x

x=1.2

==Connectedness==
How is this topic connected to something that you are interested in?
I have a heavy interest in cars. More specifically, I love race cars, and I love watching Formula1, BTCC, WRC, and other car racing series. Torque, in cars, gives you an idea of how fast the car can accelerate. But more importantly, torque can make cars do this:

[[File:burnout_loop.gif]]

and this:

[[File:corvetteburnout.gif]]

How is it connected to your major?

Torque is heavily related to mechanical engineering. Whether it be driving a conveyor belt in a factor, a driveshaft in a car, turning a wrench, or otherwise, torque is involved in almost all mechanical systems. Mechanical engineers use torque to transform energy into a useful form. For example, they use torque in electric motors to turn electric energy into rotational energy that can be used in all sorts of appliances. Mechanical engineers also take the chemical energy from gasoline combustion and turn it into torque to power planes, trains, and automobiles. Torque is applicable in all types of engineering.

Is there an interesting industrial application?

The amount of industrial applications of torque is nearly infinite, but one machine I find interesting is the lathe. The lathe uses torque and rotation to shape various materials [https://www.youtube.com/watch?v=9qt5ui3P9QA]

==History==

The idea of torque originated with Archimedes studies on levers. Archimedes may not have invented the lever, but he was one of the first scientists to investigate how they work in 241 BC.

== See also ==

[https://en.wikipedia.org/wiki/Torque]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]

===Further reading===

Books, Articles or other print media on this topic

===External links===
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
[https://en.wikipedia.org/wiki/Torque
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]

==References==

This section contains the the references you used while writing this page
[[https://en.wikipedia.org/wiki/Torque]]
[http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html]
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html]
[[Category:Angular Momentum]]

Systems with Nonzero Torque

2019-07-25T05:17:14Z

Bbreer6: /* Non-Zero Torque System using magnitude of vectors and Given Angle */

''Edited by Conner Rudzinski Fall 2018''

In certain systems, external torques have an effect on a system's angular momentum. Since these external forces do not sum to zero, we end up with a system with non-zero net torque.

==The Main Idea==

With previous systems involving torque, we've been fortunate enough to have systems where the net torque is non-zero, hence <math> \vec{L}_{final} = \vec{L}_{initial}. </math> <br>
See systems with zero net torque in "See Also" section below for more information. <br>

However, we're not always fortunate enough to have such systems. In such cases, our computations become a little more complicated, and we'll see how below.

===A Mathematical Model===

The angular momentum principle is the following: <math>{\frac{d\vec{L}}{dt}}= \vec{r} * \vec{F}_{net} = \vec{т}_{net} </math> <br>
This means that the calculation of torque is the cross product of the force and distance vectors. We'll now look at the case where <math> \vec{F}_{net} </math> is NOT equal to 0.

===How to Model in VPython===
The following code should be self explanatory and can be used as a template for modeling a system involving torque.
Important things to note are the use of cross() to calculate the cross product between two vectors and sphere.rotate() to rotate a sphere object around some axis at some angle. One could write their own sub-routine for the cross product and/or rotation, but since vPython comes with these routines, it is advantageous to exploit them.
# -*- coding: utf-8 -*-
from __future__ import division
from visual import *

NUM_LOOP_ITERATIONS = 5000 # Arbitrarily chose 5000
wheel = sphere(pos = vector(0, 0, 0), radius = 10, color = color.cyan, mass = 5)
axisOfRotation = vector(5, 0, 0) # Axis of rotation of system
force = vector(5, 0, 0) # Force acting on system
delta_t = 1
t = 0
angularMomentum= vector(20, 0, 0) # Initial angular momentum
omega = 40 # Initial angular speed
inertia = (wheel.mass * wheel.radius ** 2)/12 # Calculating intertia; ML^2 / 12
dtheta = 0
while t < 5000:
rate(500)
torque = cross(wheel.pos, force) # torque = position x force
angularMomentum += torque * delta_t # Update angular momentum
omega = angularMomentum / inertia
omegaScalar = dot(omega, norm(axisOfRotation))
dtheta += omegaScalar * delta_t
wheel.rotate(angle=dtheta, axis = axisOfRotation, origin = wheel.pos)
t += delta_t

==Examples==
===Simple===
If a constant net torque (non-zero) is exerted on an object, which of the following quantities cannot be constant? <br>
A) Moment of inertia <br>
B) Center of mass <br>
C) Angular momentum <br>
D) Angular velocity <br>
E) Angular acceleration <br>
<br>
Solution/Explanation: <br>
C, D. Why? <br>
A) Moment of inertia does not change depending on whether torque is exerted on an object; moment of inertia depends on the object and axis of rotation, nothing more. <br>
B) Center of mass doesn't change with applied torque as well. <br>
C) Angular momentum is equal to inertia times angular speed, and we establish below that angular velocity changes. <br>
D) Angular velocity changes since a constant force is being applied to the object, so it's speed/velocity must inherently increase. <br>
E) Because the torque being applied is constant, angular acceleration does not change (remember, acceleration is a measure of the rate of change of velocity!). <br>

===System with Non-Zero Torque Calculation===

''Calculate the torque of a system with a distance vector <2,1,3> and a force vector of <4,3,5>.'' <br>
This is a system with non-zero torque, so we will use the formula <math> \vec{т}_{net} = \vec{r} * \vec{F}_{net}</math>
The cross product of <math>\vec{r} * \vec{F}_{net} </math> is <math>\vec{2,1,3} * \vec{4,3,5} </math>, creating the vector <math> \vec{-4,2,2} </math>. This is the torque vector.

=== Non-Zero Torque System using magnitude of vectors and Given Angle===
''Calculate the torque of a system with a distance magnitude of 5 meters and a force magnitude of 7 newtons, working at an angle of 30 degrees.'' <br>
We will use the formula <math> \vec{т}_{net} = \vec{r} * \vec{F}_{net} * sin(theta)</math>
This calculation becomes 5*7*sin(30) and equals 17.5 Nm.

It is important to recognize that the computation involves a cross product, meaning it is not simply the multiplication of the magnitudes of the two vectors but rather the multiplication of the magnitudes of the two vectors AND the sine of the angle between them. The angle between the two vectors tells us information about how significant the lever arm is in terms of amplifying or decreasing the effect of a force, i.e., leverage.

===Middling===

[[File:ProblemAndSolution.jpg]]

==Connectedness==
#How is this topic connected to something that you are interested in?
:::I had taken physics in high school, and a majority of the concepts covered in this class were familiar to me. However, I had never seen any of the material we covered in the last unit (like angular momentum, conservation of angular momentum, torque, etc.). This material was new and something I found interesting since it went a step further with the information we learned throughout the semester.
[[File:HaloGameplay.jpg|420px|thumb|right|Example of physics being used to model the motion of vehicles in Halo 2]]
#How is it connected to your major? Is there an interesting industrial application?
:::My major is Computer Science, so I cannot really draw a clear line between my current coursework and this topic. That being said, physics is, of course, used in computer science. A good example would be in programming games where most interactions between objects involve physics of some kind, and the programmers/game designers want to model realistic situations in game. <br>
:::Another example that never really dawned upon me until about halfway through the semester was that both topics require a decent level skill in problem solving/reasoning. I've liked physics this semester because (and I guess this just applies to math in general) it's almost beautiful just to see how, in the end, everything works out and makes sense. Throughout the semester, I've never had a moment during class where I've flat out said, "No, that doesn't make sense" simply because that's never happened. I've never been a huge fan of physics, but even I can appreciate something that comes together so well.

==History==

Torque is a calculation of the rotational force applied to the system.
This concept doesn't have it's own history since it's just a section under torque, so refer to http://www.physicsbook.gatech.edu/Torque#History for more information.

== See also ==

A general description of torque:
http://www.physicsbook.gatech.edu/Torque

===External links===
[https://www.youtube.com/watch?v=mvzSjRFQbHE A brief overview on systems with non-zero torque]

==References==

Matters and Interactions: 4th Edition <br>
[https://drive.google.com/file/d/0B6hjEAwn8lB-WURaNmRvVGFjUnM/edit College Physics: Ninth Edition] <br>
[[Category:Angular Momentum]]

Torque vs Work

2019-07-25T05:03:55Z

Bbreer6: /* Example Problems */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r}_cm X \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F}*\vec{r} </math>

<i>where '''W''' is the scaler dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scaler value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = ∆K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} X \vec{F} = <2, 0, 0> m X <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces:

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

For these types of problems, it is best to split the problem into smaller pieces, identify the lever arms where torques are applied, and work your way back up.

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Gyroscopes

2019-07-25T04:51:51Z

Bbreer6: /* Connectedness */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

<math>L_{rot,r} = Iω</math>

Where the rotational angular momentum points horizontal to the rotor.

The <math>L_{rot,r}</math> will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

<math>Ω = \frac{V_{cm}}{r}</math>

Where <math>r</math> is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''<math>L_{support}</math> = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

<math>L_{rot}Ω</math>

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

<math>Ω = \frac{τCM}{L_{rot}} = \frac{rMg}{Iω}</math>

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments. At their core, gyroscopes are responsible for measuring or maintaining rotational motion, and it is obvious to see how this would be related to aviation, industrial applications, manufacturing, and vehicle development.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Gyroscopes

2019-07-25T04:38:44Z

Bbreer6: /* A Mathematical Model */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

<math>L_{rot,r} = Iω</math>

Where the rotational angular momentum points horizontal to the rotor.

The <math>L_{rot,r}</math> will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

<math>Ω = \frac{V_{cm}}{r}</math>

Where <math>r</math> is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''<math>L_{support}</math> = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

<math>L_{rot}Ω</math>

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

<math>Ω = \frac{τCM}{L_{rot}} = \frac{rMg}{Iω}</math>

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Gyroscopes

2019-07-25T04:36:17Z

Bbreer6: /* A Mathematical Model */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

<math>Ω = \frac{τCM}{L_{rot}} = \frac{rMg}{Iω}</math>

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Gyroscopes

2019-07-25T04:35:18Z

Bbreer6: /* A Mathematical Model */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

<math>Ω = \frac{τCM}{L_{rot}} = ''r''Mg/''I''ω</math>

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Torque vs Work

2019-07-15T03:18:05Z

Bbreer6: /* Mathematical Model: The Concepts */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r}_cm X \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F}*\vec{r} </math>

<i>where '''W''' is the scaler dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scaler value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = ∆K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scalar measurement, while torque is a vector representation. The two quantities must be measured individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} X \vec{F} = <2, 0, 0> m X <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

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

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces.

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Torque vs Work

2019-07-15T03:06:12Z

Bbreer6: /* Mathematical Model: The Concepts */

[[File:Wvt003.jpg|300px|thumb|right]]

When it comes to our universe, there still remains much that we do not understand. That we we do know and understand, we measure in relation to other ideas or subjects, in terms of abstracts we call units. In modern physics, and most mathematical applications we relate and measure matter and interactions in terms of these units. However, two such descriptions having the same units does necessarily connotate a representation of the same idea or measurement. Here, we will examine one such occurrence in modern physics, specifically the difference between torque and work when both are expressed as Newton*meters.

==The Newton-Meter --> Vector vs Scaler==

In modern physics, there a two concepts that, while describing different ideas, happen to utilize the same units, these units being:

<math> Newton * meters </math>
<math> (N * m) </math>

===Mathematical Model: The Concepts===

[[File:Wvt002.jpg|250px|thumb|left|Torque on an Object]]

The first of these concepts is '''torque''', which is a measurement of the moment that a force causes in a system. A moment is also known as a system's tendency of a force to cause rotation.

<math> \vec{\tau} = \vec{r}_cm X \vec{F} </math>

Torque can also be represented through rotational kinematics, whereby the rotational acceleration and the moment of inertia are multiplied together to equal torque:
<math> \vec{\tau} = \vec{I} \cdot \vec{\alpha}</math>

[[File:Wvt001.jpg|300px|thumb|right|Work on an Object]]

<i>where '''<math>\tau</math>''' is the vector cross product of the force vector and position vector relative to the center of mass</i>

The second concept is '''work''', which is the measurement of a force acting over a distance on a given system.

<math> W = \vec{F}*\vec{r} </math>

<i>where '''W''' is the scaler dot product of the force vector and change in position vector</i>

Both of these concepts are measured using the units of <math> Newton * meters </math> , however torque is a vector value while work is a scaler value. This is a key point to keep in mind.

Previously, in Physics, when we addressed the energy principle and energy in general, we describe the measurement of energy and change in energy using the units:

<math> Joules </math>
<math> (J) </math>

As it happens, work on a system is also equivalent to the change in kinetic energy of a system and therefore can also be expressed in joules. Positive work will add kinetic energy to a system whereas negative work will subtract kinetic energy from a system.

<math> W = ∆K = K_{final} - K_{initial} </math>

<i>where '''W''' is the work done and '''K''' is the kinetic energy of the system.</i>

So, work can be expressed in terms of joules, which means that torque can be expressed in terms of joules too, right? Well, it's not quite as straightforward as changing units. Remember, work is a scaler measurement, while torque is a vector representation. They must be measure individually because while work effects the linear momentum of a system, torque effects the angular momentum of a system. To put it in common terms, work relates to the movement of a system from one position to another while torque relates to the twisting of the system itself. So in fact, they represent very different concepts.

===Nuances in Work===

According to it's definition, work is a force acting over a given distance. So what about forces acting on an object that does not move? Let's look at an example. Take this block sitting on the earth and being acted on by a man who tries to lift it.

[[File:Wvt004.jpg|300px|middle]]

Let the mass of the block be <math> 5 kg </math> and the force of the man pulling up be <math> 20 N </math>.

We know: <math> F = mg </math> <i>where '''F''' is the force of gravity, '''m''' is the mass of the object, and '''g''' is the acceleration due to gravity</i>

Force due to man: <math> F_{man} = 20 N </math>

Force due to gravity: <math> F_{grav} = (5 kg) * (-9.8 m/(s^{2}) = -49 N </math>

Net force: <math> F_{net} = 20 N + (-49 N) = -29 N </math>

However, we know that the block will not move through the earth due to the force of the earth acting on it, and so the block will stay in place. Since the block does not move there is no change in position. Applying this to our formula for work:

Work by man: <math> W_{man} = \vec{F}_{man} * \vec{r}_{block} = <0, 20, 0> N * <0, 0, 0> m = (0*0)Nm + (20*0)Nm + (0*0)Nm = 0 Nm </math>

So even though the man applied a force to the block, no work was done because the block did not move. It a very important to realize that it doesn't matter how large the force applied to an object is, if the object does not move, no work is done.

In this situation: <math> W = ∆K = 0Nm </math>

===Nuances in Torque===

Like work, torque also has some interesting situations that are important to take note of. Consider the following example where a wheel is being acted on by a force that is directly toward the center of mass of the wheel.

[[File:Wvt005.jpg|300px|middle]]

Let the radius of the wheel be <math> 2 m </math> and the force applied be <math> 15 N </math>.

Using these values, let's take the cross product of position cross force to find the torque cause by the force on the wheel.

Torque: <math> \vec{\tau} = \vec{r}_{cm} X \vec{F} = <2, 0, 0> m X <-15, 0, 0> N </math>

<math> \vec{\tau} = (2 m * -15 N)(iXi) + (2 m * 0 N)(iXj) + (2 m * 0 N)(iXk) + (0 m * -15 N)(jXi) + (0 m * 0 N)(jXj) + (0 m * 0 N)(jXk) + (0 m * 0 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (-30 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (0 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 0> Nm </math>

So, even though a force was applied at a distance relative to the center of mass of the wheel, there was no net torque because the force was acting straight through the center of mass of the wheel. This is important to remember as any force acting through an objects center of mass may be disregarded as having any effect on the torque and therefore does not change the angular momentum of the object.

==Example Problems==

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

===Simple Torque===

If the center of mass of the spool of thread is at the origin, determine the torque caused by the force as shown below.

[[File:Wvt006.jpg|300px|middle]]

Solution:

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <0, 4, 0> m X <-2, 0, 0> N </math>

<math> \vec{\tau} = (0 m * -2 N)(iXi) + (0 m * 0 N)(iXj) + (0 m * 0 N)(iXk) + (4 m * -2 N)(jXi) + (4 m * 0 N)(jXj) + (4 m * 0 N)(jXk) + (0 m * -2 N)(kXi) + (0 m * 0 N)(kXj) + (0 m * 0 N)(kXk) </math>

<math> \vec{\tau} = (0 Nm)(0) + (0 Nm)(k) + (0 Nm)(-j) + (-8 Nm)(-k) + (0 Nm)(0) + (0 Nm)(i) + (0 Nm)(j) + (0 Nm)(-i) + (0 Nm)(0) </math>

<math> \vec{\tau} = <0, 0, 8> N*m </math>

===Middling Work===

In the system below, determine the net work that is accomplished by the two forces.

[[File:Wvt007.jpg|300px|middle]]

Solution:

<math> W = \vec{F}*\vec{r} </math>

<math> W_{1} = <30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{1} = (30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{1} = 450 Nm + 0 Nm + 0 Nm </math>

<math> W_{1} = 450 Nm </math>

<math> W_{2} = <-30, 0, 0> N * <15, 0, 0> m </math>

<math> W_{2} = (-30 N)(15 m) + (0 N)(0 m) + (0 N)(0 m) </math>

<math> W_{2} = -450 Nm + 0 Nm + 0 Nm </math>

<math> W_{2} = -450 Nm </math>

<math> W_{net} = W_{1} + W_{2} </math>

<math> W_{net} = 450 Nm - 450 Nm </math>

<math> W_{net} = 0 N*m </math>

===Difficult Torque===

In the complex system below, determine the given force is applied at point '''A'''. Determine the net torque about the origin due to the force.

[[File:Wvt008.jpg|300px|middle]]

Solution:

<math> \vec{r} = <12, -2, 6> m </math>

<math> \vec{\tau} = \vec{r}_{cm} X \vec{F} </math>

<math> \vec{\tau} = <12, -2, 6> m X <2, 1, 7> N </math>

<math> \vec{\tau} = (12 m * 2 N)(iXi) + (12 m * 1 N)(iXj) + (12 m * 7 N)(iXk) + (-2 m * 2 N)(jXi) + (-2 m * 1 N)(jXj) + (-2 m * 7 N)(jXk) + (6 m * 2 N)(kXi) + (6 m * 1 N)(kXj) + (6 m * 7 N)(kXk) </math>

<math> \vec{\tau} = (24 Nm)(0) + (12 Nm)(k) + (84 Nm)(-j) + (-4 Nm)(-k) + (-2 Nm)(0) + (-14 Nm)(i) + (12 Nm)(j) + (6 Nm)(-i) + (42 Nm)(0) </math>

<math> \vec{\tau} = <-20, -72, 16> N*m </math>

==Connectedness==

===Connected to My Own Interests?===

While I am more interested in digital and electronic designs, I love to tinker and create with mechanical devices. In order to make a device efficient and able to function without breaking, you must understand how work is done and the strain that torque can put on a device.

===Connected to Electrical Engineering?===

While torque and work are not integral parts of electrical engineering, it is very important to understand the two concepts because they do have the potential to alter the design of something I might construct. In this day and age, everything is electronic and most modern tools even use digital instructions. It's important when designing a mechanical tool to understand torque in order to determine the best avenue through which to run wire and to know how much extra give to allow to wire to move if the tools workload necessitates movement.

===Connected to Industry===

The industrial applications for both torque and work are enormous. From power tools, to vehicles, to simple machines. If one can do more wore with less energy, or apply greater torque with smaller forces, it greatly increases the workload that an individual can accomplish. So, yes, torque and work are quite important in industry.

==Origin==

===Torque===
The idea of torque first originated with Aristotle and his work with levers.

===Work===
Energy was also initially a notion put forth by Aristotle, but the concept of work was something later derived by several scientists and mathematicians, including Joule and Carnot.

== See also ==

===Further reading===

[http://www.physicsbook.gatech.edu/Work Wiki - Work]<div></div>
[http://www.physicsbook.gatech.edu/Torque Wiki - Torque]<div></div>
[http://www.physicsbook.gatech.edu/The_Energy_Principle Wiki - Energy Principle]<div></div>

===External links===
[https://www.youtube.com/watch?v=MHtAK1rMa1Y Explanation of Torque]<div></div>
[http://science360.gov/obj/video/9112c778-48e4-4b75-a09b-2f2d2404da12/science-nfl-football-torque Torque in Football - Cool!]<div></div>
[http://torquenitup.weebly.com/torque-problems.html Torque Example Problems]<div></div>

==References==

[http://www.wiley.com//college/sc/chabay/ Matter & Interactions 4th Edition]<div></div>
[http://physics.stackexchange.com/ Physics Stack Exchange]<div></div>
[https://en.wikipedia.org/wiki/Torque Wikipedia Torque]<div></div>

jderemer3

[[Category: Angular Momentum]]

Gyroscopes

2019-07-15T02:54:44Z

Bbreer6: /* A Mathematical Model */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

'''Ω = \frac{τCM}{L_{rot}} = ''r''Mg/''I''ω '''

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Gyroscopes

2019-07-15T02:53:58Z

Bbreer6: /* The Main Idea */

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

'''Ω = τCM/Lrot = ''r''Mg/''I''ω '''

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes are very well understood and applied in aviation. The flight characteristics of airplanes and helicopters are both affected by gyroscopes. Some flight instruments even harness the power of gyroscopes as an integral part of their functionality.

Many airplanes have a propeller mounted on the front of it. This spinning mass on the front of the airplane is a gyroscope! In fact, pilots learn of the gyroscopic precession produced by the propeller when they are learning to take off.

In traditional tail wheel airplanes, the first step during take off is to raise the tail off of the ground and continue down the runway on the main(front) tires. When you rotate the plane and lift the tail off of the ground, you are applying a couple to the whole airplane about its lateral axis. You have a force acting forward on the top of the propeller, and you have a force acting towards the tail on the bottom of the propeller. Since the propeller is a gyroscope, these forces are applied 90 degrees later in the rotation of the propeller. From the pilots seat, the propeller rotates clockwise. Therefore, the forward force that was applied on the top of the propeller, is now applied as a forward force on the right side of the airplane. The rearward force that was applied on the bottom of the propeller is now applied to the left side of the airplane. These two forces create a couple about the vertical axis of the airplane. This couple causes the airplane to yaw (turn) turn to the left when the tail is lifted off the ground. To avoid going off the runway during this time, the pilot must step on the right rudder, keeping the nose of the airplane straight down the runway.

There are three main instruments that use gyroscopes as their source of information, the attitude indicator and the turn coordinator. The attitude indicator uses a gyroscope that is aligned with the horizon. This gyroscope is mounted on a three axis gimbal, so it stays parallel with the horizon. The gyroscope then represents an ‘artificial horizon’ which is very useful for pilots flying in the clouds. Pilots can then, instantaneously determine the pitch and bank of the aircraft. Without this instrument, it is very easy for pilots to get spatially disoriented and not know which way is up or down. Pilots are trained to trust their instruments because the human body is simply not reliable for being spatially aware without a visual reference.

The second gyroscopic instrument is called a heading indicator. This gyroscope spins vertically and is free to rotate about the vertical axis. The gyroscope is aligned facing north upon the start of the airplanes avionics system. When the airplane turns, the gyroscope still points north. This is useful for a pilot to know which direction he is pointed. It is more reliable and accurate than a magnetic compass. (Magnetic compasses have errors when the plane is turning or accelerating because they are not weighted symmetrically.)

The third gyroscopic instrument is called a turn coordinator. The gyroscope is similar to the heading indicator, only the axis of rotation of the gyroscope is not vertical, it is about thirty degrees forward from vertical. When the airplane is turning directions, the gyroscope indicates that the plane is changing directions. This is useful for pilots to do “standard rate turns.” A standard rate turn is three degrees per second. Pilots know if they are doing the standard rate or not based on the turn coordinator.

Control input for helicopters is determined by gyroscopic precession. In order to move in any direction, the force on the main rotor has to be applied 90 degrees earlier in rotation of the rotor. For instance, if you want to go forward, you have to apply a force on the left side of the helicopter.

Without understanding the effects of gyroscopic precession, helicopters wouldn't fly and planes would crash.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Systems with Nonzero Torque

2019-07-15T02:51:43Z

Bbreer6: /* How to Model in VPython */

''Edited by Conner Rudzinski Fall 2018''

In certain systems, external torques have an effect on a system's angular momentum. Since these external forces do not sum to zero, we end up with a system with non-zero net torque.

==The Main Idea==

With previous systems involving torque, we've been fortunate enough to have systems where the net torque is non-zero, hence <math> \vec{L}_{final} = \vec{L}_{initial}. </math> <br>
See systems with zero net torque in "See Also" section below for more information. <br>

However, we're not always fortunate enough to have such systems. In such cases, our computations become a little more complicated, and we'll see how below.

===A Mathematical Model===

The angular momentum principle is the following: <math>{\frac{d\vec{L}}{dt}}= \vec{r} * \vec{F}_{net} = \vec{т}_{net} </math> <br>
This means that the calculation of torque is the cross product of the force and distance vectors. We'll now look at the case where <math> \vec{F}_{net} </math> is NOT equal to 0.

===How to Model in VPython===
The following code should be self explanatory and can be used as a template for modeling a system involving torque.
Important things to note are the use of cross() to calculate the cross product between two vectors and sphere.rotate() to rotate a sphere object around some axis at some angle. One could write their own sub-routine for the cross product and/or rotation, but since vPython comes with these routines, it is advantageous to exploit them.
# -*- coding: utf-8 -*-
from __future__ import division
from visual import *

NUM_LOOP_ITERATIONS = 5000 # Arbitrarily chose 5000
wheel = sphere(pos = vector(0, 0, 0), radius = 10, color = color.cyan, mass = 5)
axisOfRotation = vector(5, 0, 0) # Axis of rotation of system
force = vector(5, 0, 0) # Force acting on system
delta_t = 1
t = 0
angularMomentum= vector(20, 0, 0) # Initial angular momentum
omega = 40 # Initial angular speed
inertia = (wheel.mass * wheel.radius ** 2)/12 # Calculating intertia; ML^2 / 12
dtheta = 0
while t < 5000:
rate(500)
torque = cross(wheel.pos, force) # torque = position x force
angularMomentum += torque * delta_t # Update angular momentum
omega = angularMomentum / inertia
omegaScalar = dot(omega, norm(axisOfRotation))
dtheta += omegaScalar * delta_t
wheel.rotate(angle=dtheta, axis = axisOfRotation, origin = wheel.pos)
t += delta_t

==Examples==
===Simple===
If a constant net torque (non-zero) is exerted on an object, which of the following quantities cannot be constant? <br>
A) Moment of inertia <br>
B) Center of mass <br>
C) Angular momentum <br>
D) Angular velocity <br>
E) Angular acceleration <br>
<br>
Solution/Explanation: <br>
C, D. Why? <br>
A) Moment of inertia does not change depending on whether torque is exerted on an object; moment of inertia depends on the object and axis of rotation, nothing more. <br>
B) Center of mass doesn't change with applied torque as well. <br>
C) Angular momentum is equal to inertia times angular speed, and we establish below that angular velocity changes. <br>
D) Angular velocity changes since a constant force is being applied to the object, so it's speed/velocity must inherently increase. <br>
E) Because the torque being applied is constant, angular acceleration does not change (remember, acceleration is a measure of the rate of change of velocity!). <br>

===System with Non-Zero Torque Calculation===

''Calculate the torque of a system with a distance vector <2,1,3> and a force vector of <4,3,5>.'' <br>
This is a system with non-zero torque, so we will use the formula <math> \vec{т}_{net} = \vec{r} * \vec{F}_{net}</math>
The cross product of <math>\vec{r} * \vec{F}_{net} </math> is <math>\vec{2,1,3} * \vec{4,3,5} </math>, creating the vector <math> \vec{-4,2,2} </math>. This is the torque vector.

=== Non-Zero Torque System using magnitude of vectors and Given Angle===
''Calculate the torque of a system with a distance magnitude of 5 meters and a force magnitude of 7 newtons, working at an angle of 30 degrees.'' <br>
We will use the formula <math> \vec{т}_{net} = \vec{r} * \vec{F}_{net} * sin(theta)</math>
This calculation becomes 5*7*sin(30) and equals 17.5 Nm.

===Middling===

[[File:ProblemAndSolution.jpg]]

==Connectedness==
#How is this topic connected to something that you are interested in?
:::I had taken physics in high school, and a majority of the concepts covered in this class were familiar to me. However, I had never seen any of the material we covered in the last unit (like angular momentum, conservation of angular momentum, torque, etc.). This material was new and something I found interesting since it went a step further with the information we learned throughout the semester.
[[File:HaloGameplay.jpg|420px|thumb|right|Example of physics being used to model the motion of vehicles in Halo 2]]
#How is it connected to your major? Is there an interesting industrial application?
:::My major is Computer Science, so I cannot really draw a clear line between my current coursework and this topic. That being said, physics is, of course, used in computer science. A good example would be in programming games where most interactions between objects involve physics of some kind, and the programmers/game designers want to model realistic situations in game. <br>
:::Another example that never really dawned upon me until about halfway through the semester was that both topics require a decent level skill in problem solving/reasoning. I've liked physics this semester because (and I guess this just applies to math in general) it's almost beautiful just to see how, in the end, everything works out and makes sense. Throughout the semester, I've never had a moment during class where I've flat out said, "No, that doesn't make sense" simply because that's never happened. I've never been a huge fan of physics, but even I can appreciate something that comes together so well.

==History==

Torque is a calculation of the rotational force applied to the system.
This concept doesn't have it's own history since it's just a section under torque, so refer to http://www.physicsbook.gatech.edu/Torque#History for more information.

== See also ==

A general description of torque:
http://www.physicsbook.gatech.edu/Torque

===External links===
[https://www.youtube.com/watch?v=mvzSjRFQbHE A brief overview on systems with non-zero torque]

==References==

Matters and Interactions: 4th Edition <br>
[https://drive.google.com/file/d/0B6hjEAwn8lB-WURaNmRvVGFjUnM/edit College Physics: Ninth Edition] <br>
[[Category:Angular Momentum]]

Systems with Zero Torque

2019-07-14T05:10:14Z

Bbreer6: /* A Mathematical Model */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + (r_net)*(delta_t) that for systems with zero torque, LA_f = LA_i.

Here are some basic formulas for Torque when a system has a net Torque of Zero...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation: [[File:torque_2.jpg]]

Make sure to pay attention to the direction on the Forces when adding to Net Force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with Zero Torque
[[File:Torque_7.JPG]]

== Computation Model ==

see corresponding section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2.An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.

Systems with Zero Torque

2019-07-14T05:09:48Z

Bbreer6: /* A Mathematical Model */

'''[[Claimed By Emma Bivings]]'''

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + (r_net)*(delta_t) that for systems with zero torque, LA_f = LA_i.

Here are some basic formulas for Torque when a system has a net Torque of Zero...
[[File:Torque_1.jpg]]

Remember to pay attention to the rotation!! [[File:torque_2.jpg]]

Pay attention to the direction on the Forces when adding to Net Force: [[File:Torque_6.JPG]]

Torque vs Work:
[[File:Torque_8.JPG]]

Finally, remember this all applies to Systems with Zero Torque
[[File:Torque_7.JPG]]

== Computation Model ==

see corresponding section in http://www.physicsbook.gatech.edu/Torque

== Examples ==

1. Suppose that a star initially had a radius about that of our Sun, 7x10^8km, and that it rotated every 26 days. What would be the period of rotation if the star collapsed to a radius of 10km?<rev>Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.</rev>

Solution:

[[File:RealStar.jpg]]

2.An arrangement encountered in disk brakes and certain types of clutches is shown here. The lower disk. The lower disk, of moment of Inertia I1, is rotating with angular velocity 1. The upper disk, with moment of inertia I2, is lowered on to the bottom disk. Friction causes the two disks to adhere, and they finally rotate with the same angular velocity. Determine the final angular velocity if the initial angular velocity of the upper disk 2 was in the opposite direction as 1.

[[File: FullSizeRender (2).jpg]]

Solution:

[[File:Star (2).jpg]]

3. A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling the horizontal plane is fired into one end of the rod. As viewed from
above, the direction of the bullet’s velocity makes an angle of 60o with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the magnitude of the bullet’s velocity just before
the impact? <ref>https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.</ref>

[[File:FullSizeRender (3).jpg]]

Solution:

[[File:FullSizeRender (4).jpg]]

4. The tendon in your foot exerts a force of magnitude 720N. Determine the Torque (magnitude and direction)of this force about the ankle joint:

[[File:Torque_3.jpg]]

Solution:
[[File:Torque_5.JPG]]

5. A woman whose weight is 530N is poised at the right end of a diving board with a length of 3.90 m. The board has negligible weight and is bolted down at the left end, while being supported 1.40m away by a fulcrum. Find the forces that the bolt and the fulcrum respectively exert on the board.
[[File:Torque_9.JPG]]

Solution:
[[File:Torque_10.JPG]]

6. A 5.0m long horizontal beam weighing 315N is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53 degrees with the horizontal, and a 545 N person is standing 1.5m from the wall. Find the force in the cable, Ft, and the force is exerted on the beam by the wall, R, if the beam is in equilibrium.

Solution: [[File:Torque_11.JPG]]

== Connectedness ==

This principle directly relates to the bio-mechanical techniques used by various types of athletes, it can thus for example be incorporated into a physical analysis of extreme sports. Athletes (figure skaters, skateboarders, etc.) are able to employ this principle to increase the rates of rotation of there bodies without having to generate any net force. Given that Ltot = Ltrans + Lrot, and assuming that a change in Ltrans is trivial, it follows that Lrotf = Lroti. More specifically ωfIf = ωiIi. This relation implies that the athlete in question should be able to either increase or reduce their angular velocity by decreasing or increasing his or her moment of inertia respectively. Generally, this is accomplished by voluntarily cutting the distance between bodily appendages and the athlete's center of mass. In the example of a figure skater for instance, the moment of inertia is decreased by bringing in the arms and legs closer to the center of the body. Additionally, the athlete might crouch down in order to further decrease the total distance from his or her body’s center of mass. As a result, an inversely proportional change in the angular velocity of the athlete’s motion will occur, causing the speed of the athletes rotation either increase or decrease. <rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>
[[File:Iceskater.jpg]]<ref>http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.</ref>

As an LMC major, I am among the very few students at Georgia Tech for whom the zero-torque system method is not immediately applicable. If instead I was any sort of engineering major whatsoever, this would surely not be the case.

An immediate industrial example of a zero-torque system is alluded to in example problem two above. The zero-torque system is a very important method of abstraction in the field of control systems engineering. In particular, this perspective is instrumental for calculating selection factors for clutching and braking systems. Though often offered as separate components, their function are often combined into a single unit. When starting or stopping, they transfer energy between an output shaft and an input shaft through the point of contact. By considering the input shaft, output shaft and engagement mechanism as a closed system, researchers are enabled to make calculations that can inform them on how to engineer systems of progressively greater efficiency in regard to the mechanical advantage unique to different type of engagement system.<rev>http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.</ref> [[File:Car_clutch.png]]<ref>https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.</ref>

== History ==

The empirical analysis of what might now be described as “zero-torque systems” by various natural philosophers pointed towards the principles of angular momentum and torque long before they were formulated in Newton’s Principia. The principle of torque was indicated as early as Archimedes (c. 287 BC - c.212 BC) who postulated the law of the lever. As written below, it essentially describes an event in which zero net torque on a system results in zero angular momentum(4).<rev>http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.</ref> Much later on in 1609, the astronomer Johannes Kepler announced his discovery that planets followed an elliptical pattern around the Sun. More specifically, he claimed to have found that “a radius vector joining any planet to the sun sweeps out equal areas in equal lengths of time.” Mathematically, this area, (½)(rvsin) is proportional to angular momentum rmvsin. It was Newton’s endeavor to find an analytical solution for Kepler’s observations that lead to the derivation of his second law, The Momentum Principle, in the late 1600s.<rev>Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.</ref>

== See also ==

http://www.physicsbook.gatech.edu/The_Moments_of_Inertia
http://www.physicsbook.gatech.edu/Systems_with_Nonzero_Torque
http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle
http://www.physicsbook.gatech.edu/Torque
http://www.physicsbook.gatech.edu/Predicting_the_Position_of_a_Rotating_System

== Further Reading ==

Matter and Interactions, 4th edition, Volume 1.

== External links ==

https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/constant-angular-momentum-when-no-net-torque

https://www.youtube.com/watch?v=whVCEIfTT0M

https://www.youtube.com/watch?v=jeB4aAVQMug

== References ==

1. Browne, Michael E. "Angular Momentum." Physics for Engineering and Science. Third ed. McGraw Hill Education, 2013. 134. Print.

2.Chabay, Ruth W., and Bruce A. Sherwood. "Angular Momentum." Matter & Interactions. 4rth ed. Vol. 1 : Modern Mechanics. Hoboken, NJ: Wiley, 2012. 434,440. Print.

3.http://www.controleng.com/single-article/selection-factors-for-clutches-amp-brakes/9fbd9f3d184754e5ced13817eff2c659.html. Retrieved December 3, 2015.

4. http://www.britannica.com/biography/Archimedes. Retrieved December 3. 2015.

5. https://www.physics.ohio-state.edu/~gan/teaching/spring99/C12.pdf.Retrieved December 3, 2015.

6. https://upload.wikimedia.org/wikipedia/commons/f/fa/Car_clutch.png. Retrieved December 3, 2015.

7.http://qctimes.com/news/local/young-ice-skaters-shine-at-u-s-figure-skating-event/image_37e22f34-2c3a-5445-84cb-54972b525ad8.html. Retrieved December 3.2015.

Torque

2019-07-12T04:29:14Z

Bbreer6: /* Units */

'''This was edited by Kirsten Reynolds (Spring 2018) (Minor edit).'''

Torque is the measure of how much a force acting on an object causes that object to rotate, creating a tendency for the object to rotate about an axis, fulcrum, or pivot. Torque is most commonly classified as "twist", rotational force, or angular force to an object and applying it to a system changes the angular momentum of the system. The effectiveness of torque depends on where the force is applied and the position at which the force acts relative to a location.

==History==

The concept of torque first originated with Archimedes focused study on levers. While he did not invent the lever, his research and work on it caused him to create the block-and-tackle pulley systems, allowing people to use the principle of leverage to lift heavy objects. Building off of this, he explained how torque comes into play with objects that are twisting or rotating around a pivot, just as a lever does around the point of rotation. Using the Law of the Lever and geometric reasoning, Archimedes developed the concept of torque.

In 1884, the term "torque" was introduced into English scientific literature by James Thomson, a notable scientist remembered for his work on the improvement of water wheels, water pumps, and turbines. Before officially introducing the name torque, the twisting or torsional motion was referred to "moment of couple" or "angular force".

==Modeling and Understanding==
===A Mathematical Model===

Torque is the cross product between the distance vector, a vector from the point of pivot (A) to the point where the force is applied, and the force vector. The force vector, <math>{\vec{F}}</math>, is defined about a particular location. <div>
<div style="text-align: center;">[[File:Torque_formula.png |150x40px]]</div>

When applying a force to an object at an angle <math>{θ}</math> to the radius, a different equation is required to capture both the force of the twist and the distance from the pivot point to the place where the force is applied. This equation finds the magnitude of torque exerted by a force, <math>{\vec{F}}</math> relative to a location (A).

<div style="text-align: center;">[[File:torquemag_formula.png]]</div>

For a purely perpendicular force with a force application at <math>{θ}=90{°}</math>, <math>sin{θ}=1</math> and the torque is r<sub>A</sub>F. For a force that is parallel to the lever arm at an angle <math>{θ}=0{°}</math>, <math>sin{θ}=0</math> and the torque is zero.

====Angular Acceleration====
Net torque on a system is also equal to the moment of inertia multiplied by the angular acceleration.
<div style="text-align: center;"><math>\sumτ = Iα</math></div>

====Angular Momentum Principle====
The equation for torque is derived from the Angular Momentum Principle, which states that torque is equal to the change in length over time. Another equation used to represent torque is

<div style="text-align: center;"><math>{Δ}{\vec{L}} = {\vec{τ}} x {Δ}t</math></div>

====Units====
The SI unit of torque is the newton meter <math>{N{·}m}</math> or joule per radian <math>{J/rad}</math>. These units are produced from the dot product of a force and the distance over which it acts.

====Addition and Subtraction====
If more than one torque acts on an object, these values can be combined to calculate the overall net torque. If the torques make the object spin in opposite directions, they should be subtracted from one another. If the individual torques make an object spin in the same direction, the values should be added together.

===Direction of the Force===
[[File:Directionofforce.png |The angle at which the force is applied on the point of rotation changes the effectiveness of the twist.]]<div></div>
When applying a force to a system, the direction of the force greatly affects the torque and alters the effectiveness of twisting. As seen, a force parallel to the handle or object using to twist another is extremely ineffective and does not produce a torque. When the force only contains a perpendicular component, it is effective at twisting an object.

<br>

===Point of Application of the Force===
[[File:pointofapplication.png|The farther away from the nut the force is applied, the more effective the twist is.]] <div></div>
The point and placement of application of the force on an object also affects who effective the torque is. The further away from the point of rotation that a force is applied, the more effective the twist is. In order to make your twisting most effective, add length to provide more leverage.

<br><br>

===Direction of Torque===
[[File:Righthandrulebar.png|300x300px]]<div></div>
Because torque is a vector quantity, it is important to determine the direction in which torque occurs. The direction of torque is perpendicular to the radius from the axis and the force being applied to the system. The right-hand rule along the axis of rotation can be used to determine the direction of torque, where torque is in the direction your thumb is pointing. Torque is in the same direction of the change in angular velocity.

Put the fingers of the right hand in the direction of the distance vector (r) and curl the fingers in the direction of the force vector (F). The direction of the torque vector will be in the direction that the thumb is pointing towards.

If the motion is counterclockwise, and the thumb is pointing "out" from the page, the direction of torque can be noted with a "bullseye" pictogram. If the motion is clockwise, and the thumb is pointing "into" the page, the direction can be noted with an "X" pictogram.
<div></div>

==Examples==
===Simple===
====Problem====
A force of 50 N is applied to a wrench that is 30 cm in length. Calculate the torque if the fore is applied perpendicular to the wrench.

[[File:exampleone.png|thumb| Problem diagram]]

====Solution====
Using Equation 1,
<math>{\vec{τ}} = {\vec{r}} x {\vec{F}}</math>, you can plug in the values given for distance from point of rotation to where the force is being applied and for force.
<div><math>{\vec{r}} = 30 cm = 0.3 m</math></div>
<div><math>{\vec{F}} = -50 N</math></div>

<div style="text-align: center;"><math>{\vec{τ}} = (-50 N) × (0.3 m)</math></div>
<div style="text-align: center;"><math>{\vec{τ}} = -15 N{·}m</math></div>

===Middling===
====Problem====
[[File:exampletwo.png]]

====Solution====
Using Equation 2, plug in the values for <math>r_A</math>, <math>F</math>, and θ.

<div style="text-align: center;"><math>|τ| = (0.35 m) × (16 N) × sin(61°)</math></div>

<div style="text-align: center;"><math>|{\vec{τ}}| = 4.9 N · m</math></div>

===Difficult===
====Problem====
[[File:examplethree.png|center]]

====Solution====

First, create a free body diagram to include all of the forces acting on the system. Using the standard coordinate system, the pivot location is A.
<div></div>
[[File:examplethreesolution.png|center]]

In order to get the net torque, calculate all of the individual torques about the location A and add them up.
<div></div>
Torque due to child 1:
<div></div>
[[File:torquechildone.png|center]]
<div></div>
The magnitude of the torque of child 1 can be found by putting the distance vector and the force vector tail to tail and applying the right hand rule. It is found that the direction of the torque is into the page in a -z direction.
<div></div>
[[File:magtorquechildone.png|center]]

<div></div>
Torque due to child 2:
<div></div>
[[File:torquechildtwo.png|center]]

<div></div>
The cross product of normal force, <math>{\vec{F}}_n</math>, and total direction vector, <math>{\vec{r}}_n,A</math>, can be used to find the net torque about the point of rotation, A. The torque due to the normal force is zero because the force acts at location A, so it can't twist the seesaw.
<div></div>
[[File:nettorqueA.png|center]]
<div></div>

The net torque of the system is:
<div></div>
[[File:finalnettorque.png|center]]

For a brief introduction on a different method of calculating cross products (using matrices and cofactor expansion), watch this video:
[https://youtu.be/-pFvtQbxA0o]
(Useful for when vectors don't happen to line up neatly on the XYZ planes).

"radius" vector in the video refers to the distance vector (r).

==Connectedness==
Torque exists almost everywhere we go and is involved in nearly everything we do. If torque didn't exist, we would only be able to do things linearly in a uniform line and there wouldn't be spin, turn, or circular motion. Actions such as turning a steering wheel or opening a bottle would be impossible without the twisting motion we call torque.

The concept of torque interests me greatly as a Chemical Engineering major because torque plays a role in most chemical processes and unit operations. Using torque for reactions can alter flow rates, create shaft work, and affect the energy balances of continuous, steady state systems.

Torque has many industrial applications in industries such as aerospace, automotive, material processing, medical, robotics, oil and gas, and assembly. It is often used in finishing off materials with operations such as polishing, grinding, and deburring. Torque sensors are often used to determine the amount of power of engines, motors, turbines, and other rotating devices and the sensors make the required torque measurements automatically on many assembly and sure machines.

== See also ==

===Further reading===

[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html Torque] <div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html Angular Momentum] <div></div>
[https://en.wikipedia.org/wiki/Torque_sensor Torque Sensors] <div></div>

===External links===
[https://www.youtube.com/watch?v=QhuJn8YBtmg Video Tutorial on Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.example.RHR.html Practice Problems]<div></div>
[http://online.unitconverterpro.com/list.php?cat=torque Units of Torque]<div></div>

==References==
Chapter 11 of [https://books.google.com/books?id=Gz4HBgAAQBAJ&pg=PA544&lpg=PA544&dq=matter+and+interactions+4th+edition+torque&source=bl&ots=ShdH7G8bcV&sig=uEQbxhpX3-UqcQf4ilXjp2reG5s&hl=en&sa=X&ved=0ahUKEwin-JTU6MTJAhWDQCYKHUoLA00Q6AEIMjAD#v=onepage&q&f=false Matter & Interactions 4th Edition]<div></div>

[http://www.mikeraugh.org/Talks/UNM-2012-LawOfTheLever.pdf Law of the Lever]<div></div>
[http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html Physics of Torque]<div></div>
[https://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html What is Torque?]<div></div>

Young's Modulus

2019-06-17T05:35:25Z

Bbreer6: /* External links */

This page discusses Young's Modulus, best described as the stiffness of a material, examples/problems involving this concept, and its numerous real-world applications.
''(Claimed by Sanjana Kapur; Fall 2017)''

==The Main Idea==

Young's Modulus is a macroscopic property of a material that measures the stiffness of a solid material. It is independent of size or weight of the material, and it will change depending on the type of material for which the Young's Modulus is being measured. The uses of Young's modulus extend to two main sets of relationships, macroscopic springs and microscopic springs. In the case of macroscopic springs, Young's modulus is a measure of the stretchiness of a solid material outside of considerations of size and shape. In microscopic strings, when a solid object is modeled as a system of balls (atoms) connected by springs (an image of which can be seen as the cover image of the physics textbook Matter and Interactions I 3rd Edition), the Young's modulus constant of the material can be used to determine the 'interatomic spring stiffness' constant Ksi of a material in order to determine the stiffness and stretchiness of interatomic bonds. The SI unit of Young's Modulus is the pascal (Pa or N/m^2 or kg. m. s^−2).

===A Mathematical Model===

Mathematically, the Young's Modulus is the ratio of the stress placed upon a material to the strain that it endures. The definition of Young's Modulus can be expressed as: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math> where <math>{F}_{T}</math> is equal to the tension force, <math>A</math> is equal to the cross sectional area, <math>ΔL</math> is equal to the change in length due to the tension force, and <math>L_o</math> is equal to the initial length of the material.

Young's Modulus can also be expressed as: <math>{Y = \frac{Ksi}{d}}</math> where Y is the Young's modulus of the material being examined, Ksi is the value that represents the interatomic bond stiffness (how much a bond between two atoms will stretch), and d is the distance from the center of one atom to the center of another atom.

===A Computational Model===

#The following program calculates the microscopic and macroscopic Young's Modulus of a material. The source of this model is: https://trinket.io/python/8f0b71cda5

<pre>
import math
element =raw_input("What element are you investigating? This will calcuate macro and microscopic Young's Modulus based on an example in Matter & Interactions 3rd Ed")
print element
print "mol=" ,6.02e23, "atoms"
ma = raw_input("What is the hanging mass in kilograms?")
print "ma=", ma,"kg"
ma=float(ma)
rho = raw_input("What is the density in kg/meters-cubed?")
print "rho=", rho, "kg/m^3"
rho=float(rho)
ama=raw_input("what is the atomic mass in kilograms?")
ama=float(ama)
print "ama=atomic mass", ama ,"kg"
mol =6.02e23
print
print "calculations"
vol=(ma/rho)
print "ma/rho=volume=",vol, "m^3"
print "total number atoms = N = rho*(mol/ama)", rho*(mol/ama)
N=rho*(mol/ama)
print "edge of 1m cube, number of atoms=", math.pow(N, 1/3)
atoms= math.pow(N, 1/3)
print
print "in one meter of atoms there are =", math.pow(N, 1/3), "atoms"
diam=1/atoms
print "the diameter of one copper atom =", diam,"m"
print
#print "one atom =", atoms/1, "meters"
print "Finding the spring constant of an interatomic bond"
print
print "data"
L =raw_input("how long is the wire in meters?")
L =float(L)
print "L=length of original wire", L, "m" #length of the wire originally
S=raw_input("what is the change in length in mm")
S=float(S)
print "S=change in length when weight is hung from the wire", S, "mm" #length of stretch
Sm=(S/1000)
print "S in meters", Sm, "m"
print
print "cacluations"
print "Ks= macro spring constant for wire=weight/distance"
M=10
print "M= mass placed on the wire =", M,"kg"
weight=M*9.8
print "weight =", M*9.8, "N"
print "Ks = Weight/change in length", (M*9.8)/S,"N/m"
Ks=(M*9.8)/S
print "NL=Number of Atoms in one chain length", L/(1/atoms), "atoms"
NL=L/(1/atoms)
w=raw_input("width of the wire in millimeters?")
w =float(w)
d=raw_input("depthof the wire in millimeters?")
d = float(d)
print "w=width of rectangular wire=", (w/1000), "m"
wm=w/1000
print "d=depth of rectangular wire=", (d/1000), "m"
dm=d/1000
A=wm*dm
print "A = area of the wire", A, "m^2"
Aa=diam*diam
print "Aa = area of an atom", Aa, "m^2"
NA=(A/Aa)
print "NA=Number of atoms in a cross-sectional area of wire",NA,"atoms"
print "Km= interatomic spring constant for copper"
print "Ks=((Km*NA)/NL)"
Km=(Ks*NL)/NA
print "Km =((Ks*NL)/NA)",Km ,"N/m"
YMmac=((weight/A)*(L/S))
print "Young's Modulus: Macroscopic = (Force tension*wire length)/(wire area*change in length due to wieght", YMmac, "Pa"
YMmicro=Km/diam
print "YMmicro=Microscopic Spring Constant/atomic diameter"
print "Young's Modulus:Microscopic", YMmicro, "Pa"
</pre>

The following image demonstrates the concept of Young's Modulus. A force causes a solid material to stretch by a constant certain amount. This relationship is named Young's Modulus and is independent of mass of the object, and varies based on material.
[[File:Youngs.jpeg]]

==Examples==

===Simple===

'''Example 1.'''
A cylinder of wood has a stress of 800 and a strain of <math>8*10^-7</math>. What is Young's modulus for wood?

First we lay out the equation for the problem:

<math>Y = \frac{stress}{strain}</math>

We then plug in using the numbers given to us.

<math>Y = \frac{800}{8*10^-7}</math>

and so Y = 10^9 N/m^2

'''Example 2.'''

The Young's Modulus of Collagen in Bone is about 6 GPa [http://www.doitpoms.ac.uk/tlplib/bones/bone_mechanical.php]. Given this, determine the force applied to a cylindrical segment of collagen of radius 1 cm and length 0.75 m to cause it to deform (stretch) 1 mm.

First, consider the equation of Young's modulus most useful for this problem:

<math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Substitute the appropriate values:

<math>6000000000 = \frac{\frac{{F}_{T}}{3.14*0.01^2}}{\frac{{0.001}}{0.75}}</math>

Solve for F:

<math>{F = 2.55*10^{10} Newtons}</math>

Does this seem reasonable?
Yes! Collagen that makes up bone is a very hard material and does not stretch easily. Making bone stretch 1 mm longitudinally (using the bone as a vertical spring system) will take a lot of force.

===Middling===

'''Example 1.'''

A flan created by Dr. Schatz has a strawberry placed on it, stretching the flan from a length of 0.15 m to 0.2
m. The flan has a cross sectional area of .01. With the knowledge that flan has a Young’s
modulus of ~ 1.6e4 in tension, what force was used to stretch the flan?

First we would write out the equation for Youngs modulus: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Next we would fill in the equation for Young's Modulus: <math>{1.6e4 = \frac{\frac{{F}_{T}}{.01}}{\frac{{.05}}{.15}}}</math>

We can then fill in every thing that we know: <math>{16000 = 300*F_T}</math>

And so,<math>{F_T}</math> is equal to 53.3!

'''Example 2.'''

A man of weight 100 kg gets onto a bungee jump ride at a carnival. He is suspended in air by one rubber band (Young's Modulus of 0.01 GPa [http://www.engineeringtoolbox.com/young-modulus-d_417.html] of diameter 5 cm and original length of 10 m. Calculate the stretch of the band when the man reaches the bottom of the ride.

First, use the appropriate Young's Modulus equation:

<math>Y = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Then, substitute the appropriate values:

<math>{1e7 = \frac{{\frac{100*9.8}{3.14*0.05^2}}}{{\frac{ΔL}{10}}}}</math>

Solve:

<math>{ΔL=.125 m}</math>

===Difficult===

'''Example 1.'''

A new cylandrical flan is created by Dr. Schatz which has an orange placed on it which has a mass of .15 kg, compressing the flan a certain amount. Knowing that the initial length of the flan was .15 m and the flan has a diameter of .2 meters. With the knowledge that flan has a Young’s modulus of ~ 2e5 in tension, what length is the flan now?

First we need to write down the equation:

First we would write out the equation for Youngs modulus: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Next, let's solve for the unknowns:
Cross sectional area:

<math>{A} = pi*r^2</math>
<math>{A} = pi*.1^2</math>
<math>{A} = .0314 meters^2</math>

Force of the orange:

<math>{F}_{T} = m*g</math>
<math>{F}_{T} = .15*-9.8</math>
<math>{F}_{T} = -1.47 Newtons</math>

Next we would write out the equation for Youngs modulus: <math>{2e5 = \frac{\frac{-1.47}{.0314}}{\frac{{ΔL}}{.15}}}</math>

We can solve the equation for : <math>{-7/ΔL = 200000}</math>

And so,<math>{ΔL }</math> is equal to -3.5e-5! It was compressed 3.5e-5 meters!

'''Example 2.'''

Using the Young's modulus of Tungsten (<math>4*10^11</math>[http://www.engineeringtoolbox.com/young-modulus-d_417.html]) determine the interatomic "spring" stiffness of Tungsten.

First, find the center-to-center distance between 2 tungsten atoms.

<math>\frac{\frac{184 grams/mole}{6.022*10^23 atoms/mol}}{d^3}=19.25 g/cm^3</math> where d is the interatomic distance between Tungsten atoms. (19.25 g/cm^3 is the density of Tungsten[http://www.tungsten.com/materials/tungsten/]).

Solving, we get:

<math>d = 2.51e-8</math>

Using the appropriate formula:

<math>{Y = \frac{Ksi}{d}}</math>

<math>{4*10^11 = \frac{Ksi}{2.51e-8}}</math>

<math>{Ksi = 10040 N/m}</math>

==Connectedness==

#1. How is this topic connected to something that you are interested in?
I am interested in engineering, and Young’s Modulus is directly relevant to this field. Young’s Modulus is used to calculate the stiffness of materials, which is useful in structural engineering applications. In high school, I partook in scientific competitions and had to construct certain structures with a partner in a time limit. The stiffness of different materials available determined whether or not they would adequately hold up other parts of the structure and allow the entire structure to remain intact. Therefore, the Young’s Modulus was directly relevant to my interests.
#2. How is it connected to your major?
I am majoring in Computer Science. If I worked for an architectural firm, I could write code to calculate the Young’s Modulus of different building materials that the firm wanted to use to construct its buildings. The code could also determine whether or not the material was safe to use for a specific building design based on the value of the Young’s Modulus.
#3. Is there an interesting industrial application?
#a. Engineers and architects use the Young’s Modulus to determine whether or not the material that they are using to construct a particular structure can withstand a certain amount of force (based on the structure they are building, i.e. a bridge vs. a skyscraper).
#b. Scientists (including physicists) use Young’s Modulus to see how strong the materials that they plan to use in experiments are. (This is especially important in experiments in which a lot of pressure or force is applied to the material in question.)
#c. Doctors and scientists, such as biomedical engineers, need to use Young’s Modulus to determine whether certain materials are appropriate to use to construct prosthetics for people (i.e. how rigid or flexible the materials will be).

#Young's modulus is connected to all solid material, and it highlights the slight impression given to everything showing that things do push down ever so slightly on stuff that seems stationary.
#Young's modulus is used all the time in civil engineering and it is often used to help determine structural integrity of certain materials when deciding on a building.
# Young's modulus has several biomedical applications in prosthetics and in human disease. It is used to determine the structural characteristics of prosthetic material used for implants. Young's modulus of tissues changes with aging and is being studied as a factor to evaluate mortality related to vascular stiffness from aging.

==History==

Young's Modulus was first developed in 1727 by the famous Leonhard Euler in Switzerland, but it was further expanded upon by Italian scientist Giordano Riccati in 1782. Finally, it was given a name by the British Scientist Thomas Young who finished work on it in the 1800s. It is used in order to discover the elasticity of solid materials and shows the stress per strain of a solid material.

Young’s Modulus is named for Thomas Young (1773-1829), an Englishman who worked as a medical practitioner. Young conducted a great deal of research on the workings of the human eye, writing papers such as “The Mechanism of the Eye” and “On the Theory of Light and Colors,” which laid the foundations for what we now know about how the eye works and processes light that it receives from its surroundings. He also translated a solid portion of the demotic script of the Rosetta Stone and found a connection between the script and Egyptian hieroglyphics. He later created a dictionary filled with hieroglyphic vocabulary.

He made his famous discovery of the Young’s Modulus while working as a lecturer at the Royal Institution. He explained that the stiffness of any elasticity could be expressed as a modulus of its elasticity. He correctly calculated the stress distribution upon a bar, leading to the discovery of the Young’s Modulus formula (which is “stress” divided by “strain”). His newly discovered equation did have practical applications, as London engineers (i.e. one company was Chapman and Buhagiar) used it to construct steel columns and other building designs.

== See also ==
[http://www.physicsbook.gatech.edu/Leonhard_Euler Leonhard Euler]

[http://www.physicsbook.gatech.edu/Thomas_Young Thomas Young]

===Further reading===

The VERY in-depth wiki page which goes for beyond applications in physics 1.[https://en.wikipedia.org/wiki/Young%27s_modulus]

That resource is especially useful if you are pursuing a degree in Materials Science and Engineering (MSE) or any other related fields.

===External links===

HyperPhysics[http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html], which is a great tool for just about any entry-level physics curriculum.

==References==

Image: [http://s3.amazonaws.com/answer-board-image/48bb8d83-d333-4467-bfc0-de7c7c6d1c12.jpeg]

History: [https://en.wikipedia.org/wiki/Young%27s_modulus]

Young's Modulus Tables =
[http://www.doitpoms.ac.uk/tlplib/bones/bone_mechanical.php]
[http://www.engineeringtoolbox.com/young-modulus-d_417.html]

[[Category:Contact Interactions]]

Young's Modulus

2019-06-17T05:35:04Z

Bbreer6: /* Further reading */

This page discusses Young's Modulus, best described as the stiffness of a material, examples/problems involving this concept, and its numerous real-world applications.
''(Claimed by Sanjana Kapur; Fall 2017)''

==The Main Idea==

Young's Modulus is a macroscopic property of a material that measures the stiffness of a solid material. It is independent of size or weight of the material, and it will change depending on the type of material for which the Young's Modulus is being measured. The uses of Young's modulus extend to two main sets of relationships, macroscopic springs and microscopic springs. In the case of macroscopic springs, Young's modulus is a measure of the stretchiness of a solid material outside of considerations of size and shape. In microscopic strings, when a solid object is modeled as a system of balls (atoms) connected by springs (an image of which can be seen as the cover image of the physics textbook Matter and Interactions I 3rd Edition), the Young's modulus constant of the material can be used to determine the 'interatomic spring stiffness' constant Ksi of a material in order to determine the stiffness and stretchiness of interatomic bonds. The SI unit of Young's Modulus is the pascal (Pa or N/m^2 or kg. m. s^−2).

===A Mathematical Model===

Mathematically, the Young's Modulus is the ratio of the stress placed upon a material to the strain that it endures. The definition of Young's Modulus can be expressed as: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math> where <math>{F}_{T}</math> is equal to the tension force, <math>A</math> is equal to the cross sectional area, <math>ΔL</math> is equal to the change in length due to the tension force, and <math>L_o</math> is equal to the initial length of the material.

Young's Modulus can also be expressed as: <math>{Y = \frac{Ksi}{d}}</math> where Y is the Young's modulus of the material being examined, Ksi is the value that represents the interatomic bond stiffness (how much a bond between two atoms will stretch), and d is the distance from the center of one atom to the center of another atom.

===A Computational Model===

#The following program calculates the microscopic and macroscopic Young's Modulus of a material. The source of this model is: https://trinket.io/python/8f0b71cda5

<pre>
import math
element =raw_input("What element are you investigating? This will calcuate macro and microscopic Young's Modulus based on an example in Matter & Interactions 3rd Ed")
print element
print "mol=" ,6.02e23, "atoms"
ma = raw_input("What is the hanging mass in kilograms?")
print "ma=", ma,"kg"
ma=float(ma)
rho = raw_input("What is the density in kg/meters-cubed?")
print "rho=", rho, "kg/m^3"
rho=float(rho)
ama=raw_input("what is the atomic mass in kilograms?")
ama=float(ama)
print "ama=atomic mass", ama ,"kg"
mol =6.02e23
print
print "calculations"
vol=(ma/rho)
print "ma/rho=volume=",vol, "m^3"
print "total number atoms = N = rho*(mol/ama)", rho*(mol/ama)
N=rho*(mol/ama)
print "edge of 1m cube, number of atoms=", math.pow(N, 1/3)
atoms= math.pow(N, 1/3)
print
print "in one meter of atoms there are =", math.pow(N, 1/3), "atoms"
diam=1/atoms
print "the diameter of one copper atom =", diam,"m"
print
#print "one atom =", atoms/1, "meters"
print "Finding the spring constant of an interatomic bond"
print
print "data"
L =raw_input("how long is the wire in meters?")
L =float(L)
print "L=length of original wire", L, "m" #length of the wire originally
S=raw_input("what is the change in length in mm")
S=float(S)
print "S=change in length when weight is hung from the wire", S, "mm" #length of stretch
Sm=(S/1000)
print "S in meters", Sm, "m"
print
print "cacluations"
print "Ks= macro spring constant for wire=weight/distance"
M=10
print "M= mass placed on the wire =", M,"kg"
weight=M*9.8
print "weight =", M*9.8, "N"
print "Ks = Weight/change in length", (M*9.8)/S,"N/m"
Ks=(M*9.8)/S
print "NL=Number of Atoms in one chain length", L/(1/atoms), "atoms"
NL=L/(1/atoms)
w=raw_input("width of the wire in millimeters?")
w =float(w)
d=raw_input("depthof the wire in millimeters?")
d = float(d)
print "w=width of rectangular wire=", (w/1000), "m"
wm=w/1000
print "d=depth of rectangular wire=", (d/1000), "m"
dm=d/1000
A=wm*dm
print "A = area of the wire", A, "m^2"
Aa=diam*diam
print "Aa = area of an atom", Aa, "m^2"
NA=(A/Aa)
print "NA=Number of atoms in a cross-sectional area of wire",NA,"atoms"
print "Km= interatomic spring constant for copper"
print "Ks=((Km*NA)/NL)"
Km=(Ks*NL)/NA
print "Km =((Ks*NL)/NA)",Km ,"N/m"
YMmac=((weight/A)*(L/S))
print "Young's Modulus: Macroscopic = (Force tension*wire length)/(wire area*change in length due to wieght", YMmac, "Pa"
YMmicro=Km/diam
print "YMmicro=Microscopic Spring Constant/atomic diameter"
print "Young's Modulus:Microscopic", YMmicro, "Pa"
</pre>

The following image demonstrates the concept of Young's Modulus. A force causes a solid material to stretch by a constant certain amount. This relationship is named Young's Modulus and is independent of mass of the object, and varies based on material.
[[File:Youngs.jpeg]]

==Examples==

===Simple===

'''Example 1.'''
A cylinder of wood has a stress of 800 and a strain of <math>8*10^-7</math>. What is Young's modulus for wood?

First we lay out the equation for the problem:

<math>Y = \frac{stress}{strain}</math>

We then plug in using the numbers given to us.

<math>Y = \frac{800}{8*10^-7}</math>

and so Y = 10^9 N/m^2

'''Example 2.'''

The Young's Modulus of Collagen in Bone is about 6 GPa [http://www.doitpoms.ac.uk/tlplib/bones/bone_mechanical.php]. Given this, determine the force applied to a cylindrical segment of collagen of radius 1 cm and length 0.75 m to cause it to deform (stretch) 1 mm.

First, consider the equation of Young's modulus most useful for this problem:

<math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Substitute the appropriate values:

<math>6000000000 = \frac{\frac{{F}_{T}}{3.14*0.01^2}}{\frac{{0.001}}{0.75}}</math>

Solve for F:

<math>{F = 2.55*10^{10} Newtons}</math>

Does this seem reasonable?
Yes! Collagen that makes up bone is a very hard material and does not stretch easily. Making bone stretch 1 mm longitudinally (using the bone as a vertical spring system) will take a lot of force.

===Middling===

'''Example 1.'''

A flan created by Dr. Schatz has a strawberry placed on it, stretching the flan from a length of 0.15 m to 0.2
m. The flan has a cross sectional area of .01. With the knowledge that flan has a Young’s
modulus of ~ 1.6e4 in tension, what force was used to stretch the flan?

First we would write out the equation for Youngs modulus: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Next we would fill in the equation for Young's Modulus: <math>{1.6e4 = \frac{\frac{{F}_{T}}{.01}}{\frac{{.05}}{.15}}}</math>

We can then fill in every thing that we know: <math>{16000 = 300*F_T}</math>

And so,<math>{F_T}</math> is equal to 53.3!

'''Example 2.'''

A man of weight 100 kg gets onto a bungee jump ride at a carnival. He is suspended in air by one rubber band (Young's Modulus of 0.01 GPa [http://www.engineeringtoolbox.com/young-modulus-d_417.html] of diameter 5 cm and original length of 10 m. Calculate the stretch of the band when the man reaches the bottom of the ride.

First, use the appropriate Young's Modulus equation:

<math>Y = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Then, substitute the appropriate values:

<math>{1e7 = \frac{{\frac{100*9.8}{3.14*0.05^2}}}{{\frac{ΔL}{10}}}}</math>

Solve:

<math>{ΔL=.125 m}</math>

===Difficult===

'''Example 1.'''

A new cylandrical flan is created by Dr. Schatz which has an orange placed on it which has a mass of .15 kg, compressing the flan a certain amount. Knowing that the initial length of the flan was .15 m and the flan has a diameter of .2 meters. With the knowledge that flan has a Young’s modulus of ~ 2e5 in tension, what length is the flan now?

First we need to write down the equation:

First we would write out the equation for Youngs modulus: <math>{Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}}</math>

Next, let's solve for the unknowns:
Cross sectional area:

<math>{A} = pi*r^2</math>
<math>{A} = pi*.1^2</math>
<math>{A} = .0314 meters^2</math>

Force of the orange:

<math>{F}_{T} = m*g</math>
<math>{F}_{T} = .15*-9.8</math>
<math>{F}_{T} = -1.47 Newtons</math>

Next we would write out the equation for Youngs modulus: <math>{2e5 = \frac{\frac{-1.47}{.0314}}{\frac{{ΔL}}{.15}}}</math>

We can solve the equation for : <math>{-7/ΔL = 200000}</math>

And so,<math>{ΔL }</math> is equal to -3.5e-5! It was compressed 3.5e-5 meters!

'''Example 2.'''

Using the Young's modulus of Tungsten (<math>4*10^11</math>[http://www.engineeringtoolbox.com/young-modulus-d_417.html]) determine the interatomic "spring" stiffness of Tungsten.

First, find the center-to-center distance between 2 tungsten atoms.

<math>\frac{\frac{184 grams/mole}{6.022*10^23 atoms/mol}}{d^3}=19.25 g/cm^3</math> where d is the interatomic distance between Tungsten atoms. (19.25 g/cm^3 is the density of Tungsten[http://www.tungsten.com/materials/tungsten/]).

Solving, we get:

<math>d = 2.51e-8</math>

Using the appropriate formula:

<math>{Y = \frac{Ksi}{d}}</math>

<math>{4*10^11 = \frac{Ksi}{2.51e-8}}</math>

<math>{Ksi = 10040 N/m}</math>

==Connectedness==

#1. How is this topic connected to something that you are interested in?
I am interested in engineering, and Young’s Modulus is directly relevant to this field. Young’s Modulus is used to calculate the stiffness of materials, which is useful in structural engineering applications. In high school, I partook in scientific competitions and had to construct certain structures with a partner in a time limit. The stiffness of different materials available determined whether or not they would adequately hold up other parts of the structure and allow the entire structure to remain intact. Therefore, the Young’s Modulus was directly relevant to my interests.
#2. How is it connected to your major?
I am majoring in Computer Science. If I worked for an architectural firm, I could write code to calculate the Young’s Modulus of different building materials that the firm wanted to use to construct its buildings. The code could also determine whether or not the material was safe to use for a specific building design based on the value of the Young’s Modulus.
#3. Is there an interesting industrial application?
#a. Engineers and architects use the Young’s Modulus to determine whether or not the material that they are using to construct a particular structure can withstand a certain amount of force (based on the structure they are building, i.e. a bridge vs. a skyscraper).
#b. Scientists (including physicists) use Young’s Modulus to see how strong the materials that they plan to use in experiments are. (This is especially important in experiments in which a lot of pressure or force is applied to the material in question.)
#c. Doctors and scientists, such as biomedical engineers, need to use Young’s Modulus to determine whether certain materials are appropriate to use to construct prosthetics for people (i.e. how rigid or flexible the materials will be).

#Young's modulus is connected to all solid material, and it highlights the slight impression given to everything showing that things do push down ever so slightly on stuff that seems stationary.
#Young's modulus is used all the time in civil engineering and it is often used to help determine structural integrity of certain materials when deciding on a building.
# Young's modulus has several biomedical applications in prosthetics and in human disease. It is used to determine the structural characteristics of prosthetic material used for implants. Young's modulus of tissues changes with aging and is being studied as a factor to evaluate mortality related to vascular stiffness from aging.

==History==

Young's Modulus was first developed in 1727 by the famous Leonhard Euler in Switzerland, but it was further expanded upon by Italian scientist Giordano Riccati in 1782. Finally, it was given a name by the British Scientist Thomas Young who finished work on it in the 1800s. It is used in order to discover the elasticity of solid materials and shows the stress per strain of a solid material.

Young’s Modulus is named for Thomas Young (1773-1829), an Englishman who worked as a medical practitioner. Young conducted a great deal of research on the workings of the human eye, writing papers such as “The Mechanism of the Eye” and “On the Theory of Light and Colors,” which laid the foundations for what we now know about how the eye works and processes light that it receives from its surroundings. He also translated a solid portion of the demotic script of the Rosetta Stone and found a connection between the script and Egyptian hieroglyphics. He later created a dictionary filled with hieroglyphic vocabulary.

He made his famous discovery of the Young’s Modulus while working as a lecturer at the Royal Institution. He explained that the stiffness of any elasticity could be expressed as a modulus of its elasticity. He correctly calculated the stress distribution upon a bar, leading to the discovery of the Young’s Modulus formula (which is “stress” divided by “strain”). His newly discovered equation did have practical applications, as London engineers (i.e. one company was Chapman and Buhagiar) used it to construct steel columns and other building designs.

== See also ==
[http://www.physicsbook.gatech.edu/Leonhard_Euler Leonhard Euler]

[http://www.physicsbook.gatech.edu/Thomas_Young Thomas Young]

===Further reading===

The VERY in-depth wiki page which goes for beyond applications in physics 1.[https://en.wikipedia.org/wiki/Young%27s_modulus]

That resource is especially useful if you are pursuing a degree in Materials Science and Engineering (MSE) or any other related fields.

===External links===

HyperPhysics[http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html] which is a great tool for just about any entry level physics.

==References==

Image: [http://s3.amazonaws.com/answer-board-image/48bb8d83-d333-4467-bfc0-de7c7c6d1c12.jpeg]

History: [https://en.wikipedia.org/wiki/Young%27s_modulus]

Young's Modulus Tables =
[http://www.doitpoms.ac.uk/tlplib/bones/bone_mechanical.php]
[http://www.engineeringtoolbox.com/young-modulus-d_417.html]

[[Category:Contact Interactions]]

Change of State

2019-06-17T05:33:26Z

Bbreer6: /* History */

Created and Edited by Maite Marin-Mera (Spring 2017)
Claimed by Emma Flynn (Fall 2017)

Short Description of Topic

==The Main Idea==

All matter can move from one state to another under the right conditions. Depending on the properties of the matter changing states may require extreme temperature or pressure however it can be done. There are five different states of matter; gas, liquid, solid, plasma, and Bose-Einstein condensate.[http://www.livescience.com/46506-states-of-matter.html] The main idea of this wiki page is to discuss the properties of matter as it transitions between different states and how this relates to energy transfer.

'''An Overview'''

All matter can transition between the states dependent on its intrinsic properties. During these transitions there is a large change on the microscopic and macroscopic level of the matter. There is also typically a transfer of energy either into of from the matter undergoing the change.

'''Solid/Liquid'''

A very common phase change is between liquid and solids. This change of state is referred to as ''freezing'' (liquid to solid) or ''melting/fusion'' (solid to liquid).

''So what is going on a microscopic level?''

In a solid the atoms and molecules are packed tightly together. This tightly packed arrangement does not allow for much movement between the particles. Therefore a solid has low kinetic energy. In the liquid phase the particles of a substance have more kinetic energy that those in a solid. The atoms and molecules have more movement resulting in a higher kinetic energy.
In the change of state from solid to liquid there is energy required to overcome the binding forces that maintain its solid structure. This energy is called the heat of fusion.
In the change of state from liquid to solid energy is given off. The energy given off by this transition is the same amount as the energy required to freeze the matter.

'''Liquid/Gas'''

A very common phase change is between liquid and gases. This change of state is referred to as ''vaporization/boiling'' (liquid to gas) or ''condensation'' (gas to liquid).

''So what is going on a microscopic level?''

In a liquid the atoms and molecules are moving less than they would in the gas state. Therefore the gaseous state has a higher kinetic energy than the liquid state. This is due to the fact that atoms and molecules in liquids are still packed together more closely than the atoms and molecules in a gas.
In the change of state from liquid to gas there is energy required to overcome the bonds between the more closely packed atoms and molecules. This energy is called the heat of vaporization.
In the change of state from gas to liquid energy is given off by the transition. This energy is equal in magnitude to the energy required to transition from liquid to gas.

'''Other States'''

Transitions into the two lesser known states are much harder to analyze and understand. As these states are only present under extreme and unique conditions they will not be discussed in depth. There are links below for further reading on these topics.

'''Heating Curve'''

A common form of depicting the temperature at which a substance changes states and also how much heat is required to change state is a heating curve. The heating curve for water is shown below.

[[File:Heating_Curve.jpg]]

Note that at the point in which the liquid changes state there is no change in temperature. This is because the heat applied is going towards changing the bond structure of the matter and not towards heating the substance.

'''Phase Diagram'''

A common form of depicting the relationship between pressure, temperature, and the state of a substance is in a phase diagram. The phase diagram for water is shown below.

[[File:Phase_Diagram.jpg]]

Note that there is the boundary lines which denote and which temperature and pressure changes in state will occur.

===A Mathematical Model===

There are multiple different ways of modeling changes in state mathematically. The most common form is using the equations

''' <math> Q = m \cdot c \cdot \Delta T </math> and <math> Q = m \cdot H </math> (Heat of Fusion/Vaporization)'''

m = mass of substance
n = moles of substance
c = specific heat of substance

As shown in the heating curve above, these equations are used interchangeably depending on what the final and initial temperature the substance will be at.

===A Computational Model===

The concept of state change is a complex and interesting idea that is fundamental to a specialized field of physics known as statistical mechanics. In this field, the central model that describes a general mechanism of state change on the particle level is the Ising Model which utilizes Monte Carlo sampling of particles at certain temperatures in a distribution and assigns these particles a spin of either +1 or -1. The Ising Model is a simple, yet descriptive, simulation of energy state configurations in an “n by n” lattice of spins, with each “spin” being pointed in upwards (+1) or downwards (-1). The direction of the spin indicates a unique energy state of that spin.

Below is a sample simulation created in MATLAB that plots energy dependencies on normalized temperature. To run the simulation, you will need to download the files via the link provided below containing all source code. You will also need MATLAB in order to run the simulation. Make sure to only run the final_program.m file! It relies on the multiple helper functions that are associated with the Ising Model program (i.e. there is no need to run the other helper functions individually. ONLY run the final_program.m file!).

[https://drive.google.com/drive/folders/1ynOsjdaCjinBkIlCYk1XF5haeyKCrMta?usp=sharing]

Source: MATLAB (MathWorks)

If you run the simulation, the output should look similar to this plot:

[[Image:Ising_output.JPG |950px]]

While the output will be next to identical, there are actually slight differences in the actual values of temperature and energy of the species simulated. This is because Monte Carlo sampling was used, so each run is fed with randomly selected spins/temperatures and thus is independent of any other run.

==Examples==

===Simple===
'''1) What is the heat needed to melt 3g of ice?'''

<math> Q = m \cdot H </math>
Heat of Fusion for water= 334 J/g°C [http://www.kentchemistry.com/links/Energy/HeatFusion.htm]
Q= 3(334) = 1002 J

===Middling===
'''1) Calculate the heat needed/given off when 20g of water at 52°C is cooled to 27°C'''

Q=mcΔT
Specific Heat of Water= 4.184J/g°C [https://water.usgs.gov/edu/heat-capacity.html]

Q= 20(4.184)(27-52)= -2092 J
This means that 2092 J are given off by this process.

===Difficult===
'''1) Calculate the heat needed to heat 30 g of water from -1°C to 78°C'''

You have to break this problem into multiple different steps.

First you calculate the Q from the change in temperature to get it to 0°C.
Q1=mcΔT
Q1=30(4.184)(0-(-1))= 125.52J

Then you calculate the Q from the phase change
Q2=mH
Q2= 30(334) = 10020 J

First you calculate the Q from the change in temperature to get it to 78°C.
Q3=mcΔT
Q3=30(4.184)(78-0)= 9790.56J

Then you add all the Q's together
Q1 +Q2 +Q3= 125.52+10020+9790.56= 19936.08

==Connectedness==

This topic is related to all aspects of everyday life, phase changes are present in multiple different engineering processes. It is important to know where the energy is flowing in these changes of state in order to maximize efficiency. This is applicable to my major as a Biomedical Engineer because there are multiple processes which have changes of state and in order to keep a system balanced there has to be an influx or outflow of energy.

==History==

Changes of states of matter have been studied from the very beginning of the studies of science. From cooking to food and beverage preservation to the hydrological cycle, change of state is a pivotal concept in physics, especially when considering its implications for many related fields. The relation of state change to temperature, or average kinetic energy, is an important one with many historical examples. For instance, the failure of NASA managers to consider the temperature limits of the rubber O-rings on the solid rocket boosters when covered with water in sub-freezing temperatures led to the escape of super-heated gas that inevitably ignited the external tank, leading to the death of all seven astronauts on board the Space Shuttle Challenger. Although a fairly simple concept, the applications of the science behind state change go far beyond the surface and, in many cases, are crucial to the success of projects such as the space shuttle program, one that had to deal with environments (like entry to space/LEO) where state change was common, frequent, and hard to control/adapt to.

===Further reading===

More on a mathematical model
http://www.math.utk.edu/~vasili/475/Handouts/3.PhChgbk.1+title.pdf

===External links===

https://ch301.cm.utexas.edu/section2.php?target=thermo/enthalpy/heat-curves.html

https://craigssenseofwonder.wordpress.com/tag/phase-diagram/

http://www.chem4kids.com/files/matter_changes.html

http://www.livescience.com/46506-states-of-matter.html

http://www.ck12.org/chemistry/Multi-Step-Problems-with-Changes-of-State/lesson/Multi-Step-Problems-with-Changes-of-State-CHEM/

==References==

[[Category:Which Category did you place this in?]]

Change of State

2019-06-02T06:09:41Z

Bbreer6: /* A Computational Model */

Created and Edited by Maite Marin-Mera (Spring 2017)
Claimed by Emma Flynn (Fall 2017)

Short Description of Topic

==The Main Idea==

All matter can move from one state to another under the right conditions. Depending on the properties of the matter changing states may require extreme temperature or pressure however it can be done. There are five different states of matter; gas, liquid, solid, plasma, and Bose-Einstein condensate.[http://www.livescience.com/46506-states-of-matter.html] The main idea of this wiki page is to discuss the properties of matter as it transitions between different states and how this relates to energy transfer.

'''An Overview'''

All matter can transition between the states dependent on its intrinsic properties. During these transitions there is a large change on the microscopic and macroscopic level of the matter. There is also typically a transfer of energy either into of from the matter undergoing the change.

'''Solid/Liquid'''

A very common phase change is between liquid and solids. This change of state is referred to as ''freezing'' (liquid to solid) or ''melting/fusion'' (solid to liquid).

''So what is going on a microscopic level?''

In a solid the atoms and molecules are packed tightly together. This tightly packed arrangement does not allow for much movement between the particles. Therefore a solid has low kinetic energy. In the liquid phase the particles of a substance have more kinetic energy that those in a solid. The atoms and molecules have more movement resulting in a higher kinetic energy.
In the change of state from solid to liquid there is energy required to overcome the binding forces that maintain its solid structure. This energy is called the heat of fusion.
In the change of state from liquid to solid energy is given off. The energy given off by this transition is the same amount as the energy required to freeze the matter.

'''Liquid/Gas'''

A very common phase change is between liquid and gases. This change of state is referred to as ''vaporization/boiling'' (liquid to gas) or ''condensation'' (gas to liquid).

''So what is going on a microscopic level?''

In a liquid the atoms and molecules are moving less than they would in the gas state. Therefore the gaseous state has a higher kinetic energy than the liquid state. This is due to the fact that atoms and molecules in liquids are still packed together more closely than the atoms and molecules in a gas.
In the change of state from liquid to gas there is energy required to overcome the bonds between the more closely packed atoms and molecules. This energy is called the heat of vaporization.
In the change of state from gas to liquid energy is given off by the transition. This energy is equal in magnitude to the energy required to transition from liquid to gas.

'''Other States'''

Transitions into the two lesser known states are much harder to analyze and understand. As these states are only present under extreme and unique conditions they will not be discussed in depth. There are links below for further reading on these topics.

'''Heating Curve'''

A common form of depicting the temperature at which a substance changes states and also how much heat is required to change state is a heating curve. The heating curve for water is shown below.

[[File:Heating_Curve.jpg]]

Note that at the point in which the liquid changes state there is no change in temperature. This is because the heat applied is going towards changing the bond structure of the matter and not towards heating the substance.

'''Phase Diagram'''

A common form of depicting the relationship between pressure, temperature, and the state of a substance is in a phase diagram. The phase diagram for water is shown below.

[[File:Phase_Diagram.jpg]]

Note that there is the boundary lines which denote and which temperature and pressure changes in state will occur.

===A Mathematical Model===

There are multiple different ways of modeling changes in state mathematically. The most common form is using the equations

''' <math> Q = m \cdot c \cdot \Delta T </math> and <math> Q = m \cdot H </math> (Heat of Fusion/Vaporization)'''

m = mass of substance
n = moles of substance
c = specific heat of substance

As shown in the heating curve above, these equations are used interchangeably depending on what the final and initial temperature the substance will be at.

===A Computational Model===

The concept of state change is a complex and interesting idea that is fundamental to a specialized field of physics known as statistical mechanics. In this field, the central model that describes a general mechanism of state change on the particle level is the Ising Model which utilizes Monte Carlo sampling of particles at certain temperatures in a distribution and assigns these particles a spin of either +1 or -1. The Ising Model is a simple, yet descriptive, simulation of energy state configurations in an “n by n” lattice of spins, with each “spin” being pointed in upwards (+1) or downwards (-1). The direction of the spin indicates a unique energy state of that spin.

Below is a sample simulation created in MATLAB that plots energy dependencies on normalized temperature. To run the simulation, you will need to download the files via the link provided below containing all source code. You will also need MATLAB in order to run the simulation. Make sure to only run the final_program.m file! It relies on the multiple helper functions that are associated with the Ising Model program (i.e. there is no need to run the other helper functions individually. ONLY run the final_program.m file!).

[https://drive.google.com/drive/folders/1ynOsjdaCjinBkIlCYk1XF5haeyKCrMta?usp=sharing]

Source: MATLAB (MathWorks)

If you run the simulation, the output should look similar to this plot:

[[Image:Ising_output.JPG |950px]]

While the output will be next to identical, there are actually slight differences in the actual values of temperature and energy of the species simulated. This is because Monte Carlo sampling was used, so each run is fed with randomly selected spins/temperatures and thus is independent of any other run.

==Examples==

===Simple===
'''1) What is the heat needed to melt 3g of ice?'''

<math> Q = m \cdot H </math>
Heat of Fusion for water= 334 J/g°C [http://www.kentchemistry.com/links/Energy/HeatFusion.htm]
Q= 3(334) = 1002 J

===Middling===
'''1) Calculate the heat needed/given off when 20g of water at 52°C is cooled to 27°C'''

Q=mcΔT
Specific Heat of Water= 4.184J/g°C [https://water.usgs.gov/edu/heat-capacity.html]

Q= 20(4.184)(27-52)= -2092 J
This means that 2092 J are given off by this process.

===Difficult===
'''1) Calculate the heat needed to heat 30 g of water from -1°C to 78°C'''

You have to break this problem into multiple different steps.

First you calculate the Q from the change in temperature to get it to 0°C.
Q1=mcΔT
Q1=30(4.184)(0-(-1))= 125.52J

Then you calculate the Q from the phase change
Q2=mH
Q2= 30(334) = 10020 J

First you calculate the Q from the change in temperature to get it to 78°C.
Q3=mcΔT
Q3=30(4.184)(78-0)= 9790.56J

Then you add all the Q's together
Q1 +Q2 +Q3= 125.52+10020+9790.56= 19936.08

==Connectedness==

This topic is related to all aspects of everyday life, phase changes are present in multiple different engineering processes. It is important to know where the energy is flowing in these changes of state in order to maximize efficiency. This is applicable to my major as a Biomedical Engineer because there are multiple processes which have changes of state and in order to keep a system balanced there has to be an influx or outflow of energy.

==History==

Changes of states of matter have been studied from the very beginning of the studies of science.

===Further reading===

More on a mathematical model
http://www.math.utk.edu/~vasili/475/Handouts/3.PhChgbk.1+title.pdf

===External links===

https://ch301.cm.utexas.edu/section2.php?target=thermo/enthalpy/heat-curves.html

https://craigssenseofwonder.wordpress.com/tag/phase-diagram/

http://www.chem4kids.com/files/matter_changes.html

http://www.livescience.com/46506-states-of-matter.html

http://www.ck12.org/chemistry/Multi-Step-Problems-with-Changes-of-State/lesson/Multi-Step-Problems-with-Changes-of-State-CHEM/

==References==

[[Category:Which Category did you place this in?]]

Change of State

2019-06-02T06:09:28Z

Bbreer6: /* A Computational Model */

Created and Edited by Maite Marin-Mera (Spring 2017)
Claimed by Emma Flynn (Fall 2017)

Short Description of Topic

==The Main Idea==

All matter can move from one state to another under the right conditions. Depending on the properties of the matter changing states may require extreme temperature or pressure however it can be done. There are five different states of matter; gas, liquid, solid, plasma, and Bose-Einstein condensate.[http://www.livescience.com/46506-states-of-matter.html] The main idea of this wiki page is to discuss the properties of matter as it transitions between different states and how this relates to energy transfer.

'''An Overview'''

All matter can transition between the states dependent on its intrinsic properties. During these transitions there is a large change on the microscopic and macroscopic level of the matter. There is also typically a transfer of energy either into of from the matter undergoing the change.

'''Solid/Liquid'''

A very common phase change is between liquid and solids. This change of state is referred to as ''freezing'' (liquid to solid) or ''melting/fusion'' (solid to liquid).

''So what is going on a microscopic level?''

In a solid the atoms and molecules are packed tightly together. This tightly packed arrangement does not allow for much movement between the particles. Therefore a solid has low kinetic energy. In the liquid phase the particles of a substance have more kinetic energy that those in a solid. The atoms and molecules have more movement resulting in a higher kinetic energy.
In the change of state from solid to liquid there is energy required to overcome the binding forces that maintain its solid structure. This energy is called the heat of fusion.
In the change of state from liquid to solid energy is given off. The energy given off by this transition is the same amount as the energy required to freeze the matter.

'''Liquid/Gas'''

A very common phase change is between liquid and gases. This change of state is referred to as ''vaporization/boiling'' (liquid to gas) or ''condensation'' (gas to liquid).

''So what is going on a microscopic level?''

In a liquid the atoms and molecules are moving less than they would in the gas state. Therefore the gaseous state has a higher kinetic energy than the liquid state. This is due to the fact that atoms and molecules in liquids are still packed together more closely than the atoms and molecules in a gas.
In the change of state from liquid to gas there is energy required to overcome the bonds between the more closely packed atoms and molecules. This energy is called the heat of vaporization.
In the change of state from gas to liquid energy is given off by the transition. This energy is equal in magnitude to the energy required to transition from liquid to gas.

'''Other States'''

Transitions into the two lesser known states are much harder to analyze and understand. As these states are only present under extreme and unique conditions they will not be discussed in depth. There are links below for further reading on these topics.

'''Heating Curve'''

A common form of depicting the temperature at which a substance changes states and also how much heat is required to change state is a heating curve. The heating curve for water is shown below.

[[File:Heating_Curve.jpg]]

Note that at the point in which the liquid changes state there is no change in temperature. This is because the heat applied is going towards changing the bond structure of the matter and not towards heating the substance.

'''Phase Diagram'''

A common form of depicting the relationship between pressure, temperature, and the state of a substance is in a phase diagram. The phase diagram for water is shown below.

[[File:Phase_Diagram.jpg]]

Note that there is the boundary lines which denote and which temperature and pressure changes in state will occur.

===A Mathematical Model===

There are multiple different ways of modeling changes in state mathematically. The most common form is using the equations

''' <math> Q = m \cdot c \cdot \Delta T </math> and <math> Q = m \cdot H </math> (Heat of Fusion/Vaporization)'''

m = mass of substance
n = moles of substance
c = specific heat of substance

As shown in the heating curve above, these equations are used interchangeably depending on what the final and initial temperature the substance will be at.

===A Computational Model===

The concept of state change is a complex and interesting idea that is fundamental to a specialized field of physics known as statistical mechanics. In this field, the central model that describes a general mechanism of state change on the particle level is the Ising Model which utilizes Monte Carlo sampling of particles at certain temperatures in a distribution and assigns these particles a spin of either +1 or -1. The Ising Model is a simple, yet descriptive, simulation of energy state configurations in an “n by n” lattice of spins, with each “spin” being pointed in upwards (+1) or downwards (-1). The direction of the spin indicates a unique energy state of that spin.

Below is a sample simulation created in MATLAB that plots energy dependencies on normalized temperature. To run the simulation, you will need to download the files via the link provided below containing all source code. You will also need MATLAB in order to run the simulation. Make sure to only run the final_program.m file! It relies on the multiple helper functions that are associated with the Ising Model program (i.e. there is no need to run the other helper functions individually. ONLY run the final_program.m file!).

[https://drive.google.com/drive/folders/1ynOsjdaCjinBkIlCYk1XF5haeyKCrMta?usp=sharing]

Source: MATLAB (MathWorks)

If you run the simulation, the output should look similar to this plot:

[[Image:Ising_output.JPG |850px]]

While the output will be next to identical, there are actually slight differences in the actual values of temperature and energy of the species simulated. This is because Monte Carlo sampling was used, so each run is fed with randomly selected spins/temperatures and thus is independent of any other run.

==Examples==

===Simple===
'''1) What is the heat needed to melt 3g of ice?'''

<math> Q = m \cdot H </math>
Heat of Fusion for water= 334 J/g°C [http://www.kentchemistry.com/links/Energy/HeatFusion.htm]
Q= 3(334) = 1002 J

===Middling===
'''1) Calculate the heat needed/given off when 20g of water at 52°C is cooled to 27°C'''

Q=mcΔT
Specific Heat of Water= 4.184J/g°C [https://water.usgs.gov/edu/heat-capacity.html]

Q= 20(4.184)(27-52)= -2092 J
This means that 2092 J are given off by this process.

===Difficult===
'''1) Calculate the heat needed to heat 30 g of water from -1°C to 78°C'''

You have to break this problem into multiple different steps.

First you calculate the Q from the change in temperature to get it to 0°C.
Q1=mcΔT
Q1=30(4.184)(0-(-1))= 125.52J

Then you calculate the Q from the phase change
Q2=mH
Q2= 30(334) = 10020 J

First you calculate the Q from the change in temperature to get it to 78°C.
Q3=mcΔT
Q3=30(4.184)(78-0)= 9790.56J

Then you add all the Q's together
Q1 +Q2 +Q3= 125.52+10020+9790.56= 19936.08

==Connectedness==

This topic is related to all aspects of everyday life, phase changes are present in multiple different engineering processes. It is important to know where the energy is flowing in these changes of state in order to maximize efficiency. This is applicable to my major as a Biomedical Engineer because there are multiple processes which have changes of state and in order to keep a system balanced there has to be an influx or outflow of energy.

==History==

Changes of states of matter have been studied from the very beginning of the studies of science.

===Further reading===

More on a mathematical model
http://www.math.utk.edu/~vasili/475/Handouts/3.PhChgbk.1+title.pdf

===External links===

https://ch301.cm.utexas.edu/section2.php?target=thermo/enthalpy/heat-curves.html

https://craigssenseofwonder.wordpress.com/tag/phase-diagram/

http://www.chem4kids.com/files/matter_changes.html

http://www.livescience.com/46506-states-of-matter.html

http://www.ck12.org/chemistry/Multi-Step-Problems-with-Changes-of-State/lesson/Multi-Step-Problems-with-Changes-of-State-CHEM/

==References==

[[Category:Which Category did you place this in?]]

Change of State

2019-06-02T06:09:20Z

Bbreer6: /* A Computational Model */

Created and Edited by Maite Marin-Mera (Spring 2017)
Claimed by Emma Flynn (Fall 2017)

Short Description of Topic

==The Main Idea==

All matter can move from one state to another under the right conditions. Depending on the properties of the matter changing states may require extreme temperature or pressure however it can be done. There are five different states of matter; gas, liquid, solid, plasma, and Bose-Einstein condensate.[http://www.livescience.com/46506-states-of-matter.html] The main idea of this wiki page is to discuss the properties of matter as it transitions between different states and how this relates to energy transfer.

'''An Overview'''

All matter can transition between the states dependent on its intrinsic properties. During these transitions there is a large change on the microscopic and macroscopic level of the matter. There is also typically a transfer of energy either into of from the matter undergoing the change.

'''Solid/Liquid'''

A very common phase change is between liquid and solids. This change of state is referred to as ''freezing'' (liquid to solid) or ''melting/fusion'' (solid to liquid).

''So what is going on a microscopic level?''

In a solid the atoms and molecules are packed tightly together. This tightly packed arrangement does not allow for much movement between the particles. Therefore a solid has low kinetic energy. In the liquid phase the particles of a substance have more kinetic energy that those in a solid. The atoms and molecules have more movement resulting in a higher kinetic energy.
In the change of state from solid to liquid there is energy required to overcome the binding forces that maintain its solid structure. This energy is called the heat of fusion.
In the change of state from liquid to solid energy is given off. The energy given off by this transition is the same amount as the energy required to freeze the matter.

'''Liquid/Gas'''

A very common phase change is between liquid and gases. This change of state is referred to as ''vaporization/boiling'' (liquid to gas) or ''condensation'' (gas to liquid).

''So what is going on a microscopic level?''

In a liquid the atoms and molecules are moving less than they would in the gas state. Therefore the gaseous state has a higher kinetic energy than the liquid state. This is due to the fact that atoms and molecules in liquids are still packed together more closely than the atoms and molecules in a gas.
In the change of state from liquid to gas there is energy required to overcome the bonds between the more closely packed atoms and molecules. This energy is called the heat of vaporization.
In the change of state from gas to liquid energy is given off by the transition. This energy is equal in magnitude to the energy required to transition from liquid to gas.

'''Other States'''

Transitions into the two lesser known states are much harder to analyze and understand. As these states are only present under extreme and unique conditions they will not be discussed in depth. There are links below for further reading on these topics.

'''Heating Curve'''

A common form of depicting the temperature at which a substance changes states and also how much heat is required to change state is a heating curve. The heating curve for water is shown below.

[[File:Heating_Curve.jpg]]

Note that at the point in which the liquid changes state there is no change in temperature. This is because the heat applied is going towards changing the bond structure of the matter and not towards heating the substance.

'''Phase Diagram'''

A common form of depicting the relationship between pressure, temperature, and the state of a substance is in a phase diagram. The phase diagram for water is shown below.

[[File:Phase_Diagram.jpg]]

Note that there is the boundary lines which denote and which temperature and pressure changes in state will occur.

===A Mathematical Model===

There are multiple different ways of modeling changes in state mathematically. The most common form is using the equations

''' <math> Q = m \cdot c \cdot \Delta T </math> and <math> Q = m \cdot H </math> (Heat of Fusion/Vaporization)'''

m = mass of substance
n = moles of substance
c = specific heat of substance

As shown in the heating curve above, these equations are used interchangeably depending on what the final and initial temperature the substance will be at.

===A Computational Model===

The concept of state change is a complex and interesting idea that is fundamental to a specialized field of physics known as statistical mechanics. In this field, the central model that describes a general mechanism of state change on the particle level is the Ising Model which utilizes Monte Carlo sampling of particles at certain temperatures in a distribution and assigns these particles a spin of either +1 or -1. The Ising Model is a simple, yet descriptive, simulation of energy state configurations in an “n by n” lattice of spins, with each “spin” being pointed in upwards (+1) or downwards (-1). The direction of the spin indicates a unique energy state of that spin.

Below is a sample simulation created in MATLAB that plots energy dependencies on normalized temperature. To run the simulation, you will need to download the files via the link provided below containing all source code. You will also need MATLAB in order to run the simulation. Make sure to only run the final_program.m file! It relies on the multiple helper functions that are associated with the Ising Model program (i.e. there is no need to run the other helper functions individually. ONLY run the final_program.m file!).

[https://drive.google.com/drive/folders/1ynOsjdaCjinBkIlCYk1XF5haeyKCrMta?usp=sharing]

Source: MATLAB (MathWorks)

If you run the simulation, the output should look similar to this plot:

[[Image:Ising_output.JPG |250px]]

While the output will be next to identical, there are actually slight differences in the actual values of temperature and energy of the species simulated. This is because Monte Carlo sampling was used, so each run is fed with randomly selected spins/temperatures and thus is independent of any other run.

==Examples==

===Simple===
'''1) What is the heat needed to melt 3g of ice?'''

<math> Q = m \cdot H </math>
Heat of Fusion for water= 334 J/g°C [http://www.kentchemistry.com/links/Energy/HeatFusion.htm]
Q= 3(334) = 1002 J

===Middling===
'''1) Calculate the heat needed/given off when 20g of water at 52°C is cooled to 27°C'''

Q=mcΔT
Specific Heat of Water= 4.184J/g°C [https://water.usgs.gov/edu/heat-capacity.html]

Q= 20(4.184)(27-52)= -2092 J
This means that 2092 J are given off by this process.

===Difficult===
'''1) Calculate the heat needed to heat 30 g of water from -1°C to 78°C'''

You have to break this problem into multiple different steps.

First you calculate the Q from the change in temperature to get it to 0°C.
Q1=mcΔT
Q1=30(4.184)(0-(-1))= 125.52J

Then you calculate the Q from the phase change
Q2=mH
Q2= 30(334) = 10020 J

First you calculate the Q from the change in temperature to get it to 78°C.
Q3=mcΔT
Q3=30(4.184)(78-0)= 9790.56J

Then you add all the Q's together
Q1 +Q2 +Q3= 125.52+10020+9790.56= 19936.08

==Connectedness==

This topic is related to all aspects of everyday life, phase changes are present in multiple different engineering processes. It is important to know where the energy is flowing in these changes of state in order to maximize efficiency. This is applicable to my major as a Biomedical Engineer because there are multiple processes which have changes of state and in order to keep a system balanced there has to be an influx or outflow of energy.

==History==

Changes of states of matter have been studied from the very beginning of the studies of science.

===Further reading===

More on a mathematical model
http://www.math.utk.edu/~vasili/475/Handouts/3.PhChgbk.1+title.pdf

===External links===

https://ch301.cm.utexas.edu/section2.php?target=thermo/enthalpy/heat-curves.html

https://craigssenseofwonder.wordpress.com/tag/phase-diagram/

http://www.chem4kids.com/files/matter_changes.html

http://www.livescience.com/46506-states-of-matter.html

http://www.ck12.org/chemistry/Multi-Step-Problems-with-Changes-of-State/lesson/Multi-Step-Problems-with-Changes-of-State-CHEM/

==References==

[[Category:Which Category did you place this in?]]