Physics Book - User contributions [en]

Systems with Zero Torque

2016-11-27T02:31:47Z

Ebivings3: /* Examples */

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

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + rnet*deltat that for systems with zero torque, LAf = LAi.

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.

Systems with Zero Torque

2016-11-27T02:09:28Z

Ebivings3: /* A Mathematical Model */

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

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + rnet*deltat that for systems with zero torque, LAf = LAi.

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]]

More things to remember: [[File:Torque_6.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]]

== 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.

2016-11-27T01:59:22Z

Ebivings3: k

Systems with Zero Torque

2016-11-27T01:58:57Z

Ebivings3: /* Examples */

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

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + rnet*deltat that for systems with zero torque, LAf = LAi.

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]]

== 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:

== 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

2016-11-27T01:58:29Z

Ebivings3: /* Examples */

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

== A Mathematical Model ==

It follows from the angular momentum principle, LAf = LAi + rnet*deltat that for systems with zero torque, LAf = LAi.

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]]

== 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]]

== 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.

2016-11-27T01:36:02Z

Ebivings3: /* Torque - Emma Bivings */

== 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 = rA x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to Fnet in regular momentum calculations. Just like how a collection of forces acting on a system is called Fnet, a collection of torques acting on a system is τnet.

===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:
*τA = rAFsinθ
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 rA 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.
===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]]

== Circuits (R, LC, RL) ==

Circuit Elements

In terms of physics, an electrical circuit is a system which is composed of certain combinations of circuit elements, including resistors, capacitors, and inductors. When used in different combinations, these elements create R, LC, RL, RC circuits. The elements that make up these circuits include capacitors, resistors, batteries, ammeters, voltmeters, ohmmeters. When an electric field is applied to a conductor, the mobile charges in the conductor experience forces, and to move in the direction of those forces. Active circuits do not experience equilibrium, rather they experience a steady state in which means that charges are moving, but their drift velocities do not change with time. For circuits, equilibriums means that there is no current flowing.

Contents [hide]
1 The Main Idea
1.1 A Mathematical Model
1.2 A Computational Model
2 Examples
2.1 Simple
2.2 Middling
2.3 Difficult
3 Connectedness
4 History
5 See also
5.1 Further reading
5.2 External links
6 References
The Main Idea
To create an active circuit there must be a specific combination of the elements above as well as a nonzero electric field in the wires. The direction of the electric field at every location must be along the wire since the current flow follows the wire. According to the loop rule, energy must be conserved along any closed path in a circuit. Although, using resistors, capacitors, ammeters, voltmeters, ohmmeters, and batteries the flow of current can change with time.

A Mathematical Model
In the steady state the net electron current can be shown in the equation:
i=nAuE

the conventional current can be solved for through the equation:
I=|q|nAuE

Energy conservation (the loop rule):
Net change in potential=0

A Computational Model
[[/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 6.45.05 PM.jpg]]
A proton moving in a circular motion parallel to the xz plane. Because the particle has both an and y component of velocity, the particles path is a helix.

Example 1

Solve for the current in the resistor:

[File:/Users/maggiegarratt/Desktop/simple.jpg]

Example 2

[File:/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 7.59.03 PM.jpg]

Example

[File:/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 7.56.37 PM.jpg]

Circuits provide the world with many conveniences. From simple circuits within flashlights to parallel circuits in household lighting everyday life is made simpler because of their existence. The difference between parallel and series circuits can be noted through the reaction of Christmas lights to a bulb burning out. If a bulb burns out in a series circuit the entire string of lights will burn out as well. This is because in a series circuit there is only one pathway for the current to flow. When the pathway is disrupted the current is in equilibrium and stops flowing. On the other hand, if a bulb burns out and the remaining bulbs stay lit you know they are connected through a parallel circuit. When the bulbs are connected in parallel there are multiple alternate pathways for the currents to travel in.

Many careers necessitate an in-depth knowledge of circuits. For the major of mechanical engineering a knowledge of circuits is beneficial for multiple applications. When designing electronic technologies it is necessary to understand the circuitry within the design.

History
Early investigations of static electricity go back hundreds of years. Static electricity is a transfer of electrons produced by friction, like when you rub a balloon across a sweater. A spark or very brief flow of current can occur when charged objects come into contact, but there is no continuous flow of current. In the absence of a continuous current, there is no useful application of electricity.
The invention of the battery -- which could produce a continuous flow of current -- made possible the development of the first electric circuits. Alessandro Volta invented the first battery, the voltaic pile, in 1800. The very first circuits used a battery and electrodes immersed in a container of water. The flow of current through the water produced hydrogen and oxygen.
The first widespread application of electric circuits for practical use was for electric lighting. Shortly after Thomas Edison invented his incandescent light bulb, he sought practical applications for it by developing an entire power generation and distribution system. The first such system in the United States was the Pearl Street Station in downtown Manhattan. It provided a few square blocks of the city with electric power, primarily for illumination.
One classification of circuits has to do with the nature of the current flow. The earliest circuits were battery-powered, which made in a steady, constant current that always flowed in the same direction. This is direct current, or DC. The use of DC continued through the time of the first electric power systems. A major problem with the DC system was that power stations could serve an area of only about a square mile because of power loss in the wires.
In 1883, engineers proposed harnessing the tremendous hydroelectric power potential of Niagara Falls to supply the needs of Buffalo, N.Y. Although this power would ultimately go beyond Buffalo to New York City and even farther, there was an initial problem with distance. Buffalo was only 16 miles from Niagara Falls, but the idea was unworkable -- until Nikola Tesla made it possible, as we'll see on the next page.

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

Further reading

The Art of ElectronicsApr 9, 2015
by Paul Horowitz and Winfield Hill

Practical Electronics for Inventors, Third Edition Jan 31, 2013
by Paul Scherz and Simon Monk

Make: Electronics: Learning Through Discovery Sep 7, 2015
by Charles Platt

External links:
https://www.khanacademy.org/science/physics/circuits-topic

https://learn.sparkfun.com/tutorials/what-is-a-circuit

http://www.allaboutcircuits.com

References:
https://www.webassign.net/ebooks/mi4/toc.html?page=23.8

http://science.howstuffworks.com/environmental/energy/circuit3.htm

Circuits

2016-10-31T19:37:40Z

Ebivings3:

== Torque - '''Emma Bivings'''==

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 = rA x F .
Applying a torque to an object changes the angular momentum of that object. Torque, in angular momentum calculations, is analogus to Fnet in regular momentum calculations. Just like how a collection of forces acting on a system is called Fnet, a collection of torques acting on a system is τnet.

===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:
*τA = rAFsinθ
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 rA 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.
===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]]

== Circuits (R, LC, RL) ==

Circuit Elements

In terms of physics, an electrical circuit is a system which is composed of certain combinations of circuit elements, including resistors, capacitors, and inductors. When used in different combinations, these elements create R, LC, RL, RC circuits. The elements that make up these circuits include capacitors, resistors, batteries, ammeters, voltmeters, ohmmeters. When an electric field is applied to a conductor, the mobile charges in the conductor experience forces, and to move in the direction of those forces. Active circuits do not experience equilibrium, rather they experience a steady state in which means that charges are moving, but their drift velocities do not change with time. For circuits, equilibriums means that there is no current flowing.

Contents [hide]
1 The Main Idea
1.1 A Mathematical Model
1.2 A Computational Model
2 Examples
2.1 Simple
2.2 Middling
2.3 Difficult
3 Connectedness
4 History
5 See also
5.1 Further reading
5.2 External links
6 References
The Main Idea
To create an active circuit there must be a specific combination of the elements above as well as a nonzero electric field in the wires. The direction of the electric field at every location must be along the wire since the current flow follows the wire. According to the loop rule, energy must be conserved along any closed path in a circuit. Although, using resistors, capacitors, ammeters, voltmeters, ohmmeters, and batteries the flow of current can change with time.

A Mathematical Model
In the steady state the net electron current can be shown in the equation:
i=nAuE

the conventional current can be solved for through the equation:
I=|q|nAuE

Energy conservation (the loop rule):
Net change in potential=0

A Computational Model
[[/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 6.45.05 PM.jpg]]
A proton moving in a circular motion parallel to the xz plane. Because the particle has both an and y component of velocity, the particles path is a helix.

Example 1

Solve for the current in the resistor:

[File:/Users/maggiegarratt/Desktop/simple.jpg]

Example 2

[File:/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 7.59.03 PM.jpg]

Example

[File:/Users/maggiegarratt/Desktop/Screen Shot 2015-12-05 at 7.56.37 PM.jpg]

Circuits provide the world with many conveniences. From simple circuits within flashlights to parallel circuits in household lighting everyday life is made simpler because of their existence. The difference between parallel and series circuits can be noted through the reaction of Christmas lights to a bulb burning out. If a bulb burns out in a series circuit the entire string of lights will burn out as well. This is because in a series circuit there is only one pathway for the current to flow. When the pathway is disrupted the current is in equilibrium and stops flowing. On the other hand, if a bulb burns out and the remaining bulbs stay lit you know they are connected through a parallel circuit. When the bulbs are connected in parallel there are multiple alternate pathways for the currents to travel in.

Many careers necessitate an in-depth knowledge of circuits. For the major of mechanical engineering a knowledge of circuits is beneficial for multiple applications. When designing electronic technologies it is necessary to understand the circuitry within the design.

History
Early investigations of static electricity go back hundreds of years. Static electricity is a transfer of electrons produced by friction, like when you rub a balloon across a sweater. A spark or very brief flow of current can occur when charged objects come into contact, but there is no continuous flow of current. In the absence of a continuous current, there is no useful application of electricity.
The invention of the battery -- which could produce a continuous flow of current -- made possible the development of the first electric circuits. Alessandro Volta invented the first battery, the voltaic pile, in 1800. The very first circuits used a battery and electrodes immersed in a container of water. The flow of current through the water produced hydrogen and oxygen.
The first widespread application of electric circuits for practical use was for electric lighting. Shortly after Thomas Edison invented his incandescent light bulb, he sought practical applications for it by developing an entire power generation and distribution system. The first such system in the United States was the Pearl Street Station in downtown Manhattan. It provided a few square blocks of the city with electric power, primarily for illumination.
One classification of circuits has to do with the nature of the current flow. The earliest circuits were battery-powered, which made in a steady, constant current that always flowed in the same direction. This is direct current, or DC. The use of DC continued through the time of the first electric power systems. A major problem with the DC system was that power stations could serve an area of only about a square mile because of power loss in the wires.
In 1883, engineers proposed harnessing the tremendous hydroelectric power potential of Niagara Falls to supply the needs of Buffalo, N.Y. Although this power would ultimately go beyond Buffalo to New York City and even farther, there was an initial problem with distance. Buffalo was only 16 miles from Niagara Falls, but the idea was unworkable -- until Nikola Tesla made it possible, as we'll see on the next page.

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

Further reading

The Art of ElectronicsApr 9, 2015
by Paul Horowitz and Winfield Hill

Practical Electronics for Inventors, Third Edition Jan 31, 2013
by Paul Scherz and Simon Monk

Make: Electronics: Learning Through Discovery Sep 7, 2015
by Charles Platt

External links:
https://www.khanacademy.org/science/physics/circuits-topic

https://learn.sparkfun.com/tutorials/what-is-a-circuit

http://www.allaboutcircuits.com

References:
https://www.webassign.net/ebooks/mi4/toc.html?page=23.8

http://science.howstuffworks.com/environmental/energy/circuit3.htm

Circuits