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Let's say you have a system of objects, possibly infinitely many. You exert certain forces on different objects in the system. No other external forces are supplied, so the only forces are internal forces between objects and forces from you. Suppose the system responds by rotating about a fixed axis (each object in the system is rotating at the same angular velocity about the axis, though this velocity need not be constant with respect to time). If you fix the forces you exert and the mass and position of the objects, then is it true that the angular velocity and direction of rotation of the system may vary? This occurs when the internal forces change. All you know about them is that they sum to $\vec{0}$, so I think it's possible for them to be chosen so that the angular velocity and direction change. As a rudimentary example, if the system consists of two objects, each with the same mass, they are rotating about the midpoint of the segment connecting their positions, and there are no external forces, then if you make the force on one object point towards the other object, you vary the magnitude of the force, and you set the initial velocities of the two objects accordingly, then the angular velocity can vary quite a bit.

Suppose the system is a uniform ball, you exert a force at a point on the surface of the ball, and the ball rotates about some axis through the center of the ball. If the answer to my question is yes, then does computing the moment of inertia of the ball relative to the axis, computing the torque generated by your force relative to the axis, and dividing their magnitudes not necessarily give the angular acceleration of the ball? If so, are there additional assumptions about the internal forces within a ball, or am I missing something?

I am fairly new to physics, and this has been bothering me recently. I apologize if I did not think this through or if I had a misunderstanding of some sort.

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  • $\begingroup$ Are you assuming the system of objects is rigid (as in the example of a uniform ball) or not? $\endgroup$ – BowlOfRed Sep 19 '16 at 7:26
  • $\begingroup$ Please tell me if the following is correct. I read the first paragraph of en.wikipedia.org/wiki/Rigid_body, and from what I understand, my system is rigid because the distance between any two objects is constant with respect to time. This is because every object is rotating about the axis at the same rate, so if you let $A$ and $B$ be the positions of two different objects in the system and $C$ be an arbitrary point on the axis of rotation, then triangle $ABC$ is the same at any point in time, so $AB$ is constant. $\endgroup$ – A Johnson Sep 19 '16 at 16:08
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If the object is rigid, then internal forces can't change the momentum or the speed of the object (either linear or angular).

While you can model the object as having certain internal forces, and you can see that those forces might create a torque, there will always be additional forces in the other direction that create an opposite torque. Altogether, the sum of torques about the center of mass from internal forces will be zero. Therefore, we usually ignore such forces.

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  • $\begingroup$ Is the following correct? The second paragraph is not necessarily true for nonrigid systems because even though the force from object $i$ onto object $j$ is the negation of the force from $j$ onto $i$, the torques generated by these forces are not necessarily negations of one another because the position vectors of $i$ and $j$ relative to the axis of rotation can be different. In the example I gave with two objects rotating about their center of mass, the position vectors were negations of each other, so the torques ended up compounding. However, in rigid bodies, the internal forces act in... $\endgroup$ – A Johnson Sep 19 '16 at 16:28
  • $\begingroup$ ...such a way that they generate no net torque. Is there a deeper reason for this, or is this the definition of rigid? $\endgroup$ – A Johnson Sep 19 '16 at 16:29

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