Predicting the Position of a Rotating System: Difference between revisions
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''(claimed by Anna Marie Whitacre, awhitacre7)'' | |||
[[File:Rotating_Sphere.gif|thumb|right|300px|Visual of a rotating system.]] | [[File:Rotating_Sphere.gif|thumb|right|300px|Visual of a rotating system.]] | ||
In order to provide a cohesive and detailed model of the motion of a rotating object ( | In order to provide a cohesive and detailed model of the motion of a rotating object (including [[Systems with Zero Torque]] and [[Systems with Nonzero Torque]]) it is necessary to predict the position of the system. | ||
==The Main Idea== | ==The Main Idea== | ||
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===A Mathematical Model=== | ===A Mathematical Model=== | ||
Given that the system is indeed rotating, the update form of [[The Angular Momentum Principle]] is applied | Given that the system in question is indeed rotating, the update form of [[The Angular Momentum Principle]] is always applied about the the center of mass in our calculations. | ||
The update form of the Angular Momentum Principle is as follows: <math>\vec{L}_{rot,f}=\vec{L}_{rot,i}+\vec{\tau}_{net}\Delta t</math> where <math>{L}</math> is rotational angular momentum and <math>\tau</math> is net torque from the surroundings. | The update form of the Angular Momentum Principle is as follows: <math>\vec{L}_{rot,f}=\vec{L}_{rot,i}+\vec{\tau}_{net}\Delta t</math> where <math>{L}</math> is rotational angular momentum and <math>\tau</math> is net torque from the surroundings. | ||
In basic situations, the update form of the Angular Momentum Principle can further be simplified to account for [[The Moments of Inertia]], <math>I</math>, and the [[Angular Velocity]], <math>\omega</math> of the rotating system. This simplification of the update form of the Angular Momentum Principle yields the following mathematical equation: <math>I\omega _{f}=I\omega _{i}+RF\Delta t</math> where net torque, <math>\vec{\tau}_{net}</math> is simplified to the product of all the forces acting on the system and the locations to which these forces are applied, <math>RF</math>. | |||
===A Computational Model=== | ===A Computational Model=== | ||
Revision as of 14:31, 1 December 2015
(claimed by Anna Marie Whitacre, awhitacre7)
In order to provide a cohesive and detailed model of the motion of a rotating object (including Systems with Zero Torque and Systems with Nonzero Torque) it is necessary to predict the position of the system.
The Main Idea
The position of a rotating system can be predicted by predicting the angle over which the object will rotate though out time. Basically, the key to finding out how much a rotating object has moved (over a specific time interval) is the angle through which it moves.
A Mathematical Model
Given that the system in question is indeed rotating, the update form of The Angular Momentum Principle is always applied about the the center of mass in our calculations.
The update form of the Angular Momentum Principle is as follows: [math]\displaystyle{ \vec{L}_{rot,f}=\vec{L}_{rot,i}+\vec{\tau}_{net}\Delta t }[/math] where [math]\displaystyle{ {L} }[/math] is rotational angular momentum and [math]\displaystyle{ \tau }[/math] is net torque from the surroundings.
In basic situations, the update form of the Angular Momentum Principle can further be simplified to account for The Moments of Inertia, [math]\displaystyle{ I }[/math], and the Angular Velocity, [math]\displaystyle{ \omega }[/math] of the rotating system. This simplification of the update form of the Angular Momentum Principle yields the following mathematical equation: [math]\displaystyle{ I\omega _{f}=I\omega _{i}+RF\Delta t }[/math] where net torque, [math]\displaystyle{ \vec{\tau}_{net} }[/math] is simplified to the product of all the forces acting on the system and the locations to which these forces are applied, [math]\displaystyle{ RF }[/math].
A Computational Model
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