Systems with Zero Torque - Revision history

Yshelar3 at 22:19, 3 December 2024

2024-12-03T22:19:18Z

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	<span style="color:blue">'''Claimed ~~and edited~~ by Yash Shelar (Fall 2024).'''</span>		<span style="color:blue">'''Claimed by Yash Shelar (Fall 2024).'''</span>

	= Systems with Zero Torque =		= Systems with Zero Torque =

Yshelar3: Made all changes in previous edit and then added my name for credit attribution

2024-12-03T22:18:57Z

Made all changes in previous edit and then added my name for credit attribution

← Older revision		Revision as of 18:18, 3 December 2024
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			<span style="color:blue">'''Claimed and edited by Yash Shelar (Fall 2024).'''</span>

	= Systems with Zero Torque =		= Systems with Zero Torque =

Yshelar3: Added more examples, reformatted outline to match template, added a computational model, redid the mathematical model and explained more about lever arm, and added to connectedness and history. Made further readings and references more robust.

2024-12-03T22:16:23Z

Added more examples, reformatted outline to match template, added a computational model, redid the mathematical model and explained more about lever arm, and added to connectedness and history. Made further readings and references more robust.

Show changes

Bbreer6: /* A Mathematical Model */

2019-07-25T17:38:02Z

A Mathematical Model

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	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>.		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>...		Here are some basic formulas for finding torque when a system has a net <b>torque of zero</b>...

Bbreer6: /* Examples */

2019-07-25T17:21:21Z

Examples

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	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~~>		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:		Solution:

Bbreer6: /* History */

2019-07-25T17:20:19Z

History

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	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>		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 ==		== See also ==

Bbreer6: /* Connectedness */

2019-07-25T17:18:11Z

Connectedness

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	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>		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>		[[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.<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>

	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 ==		== History ==

Bbreer6: /* References */

2019-07-25T17:15:07Z

References

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	<~~ref~~>		<references>

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

Bbreer6: /* References */

2019-07-25T17:14:51Z

References

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			<ref>

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

Bbreer6: /* Computation Model */

2019-07-25T17:13:18Z

Computation Model

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



	== Examples ==		== Examples ==