# Conceptual question about force/torque on a rigid object

1. Dec 2, 2014

### Agrasin

I've taken up to the equivalent of first year undergrad mechanics, but this simple concept is unclear to me.

Say a force F is applied (perpendicular) to a rod (in a vacuum with no gravity-- F is the only external force) at a non center of mass point A for a tiny time dt. How does it move?

My thoughts:

Its CM accelerates with a = F/m. A force on an object is a force on an object regardless of where-- Newton's 2nd Law applies.

There is also some rotation. That's what I don't understand well. There is rotation with respect to any point except A. Say I pick the CM. The torque applied with respect to the CM is τ = rF where r is the distance between the point A and the CM. Then the final angular momentum with respect to the CM can be found from τ = dL/dt. But say I pick a different point, like point A itself. Then r is different (in the case of point A, r = 0), and the final angular momentum is different (in this case 0) with respect to that point. Which value of L is correct? The point I choose shouldn't matter, should it?

Also, further thought makes be think that my first thought is false (it will not necessarily accelerate with a = F/m). What if F was applied at the tip of the rod? Then the rod would spin much more than it would linearly accelerate.

If you could explain how to analyze this situation with kinematics, momentum, and energy, that would be very helpful. Or at least (first) help me understand it with kinematics and momenta.

2. Dec 2, 2014

### Staff: Mentor

Your first thought is correct. The point you choose does make a difference and the final angular momentum will be different if you choose different points. Also, the final KE will be different if you choose different points. Can you think why both of those vary dependent on the point?

3. Dec 2, 2014

### Agrasin

@DaleSpam, thank you for the prompt help. Now that I think about it, I suppose angular momentum would be different between points because L is not a property of the object, it is a property of each reference point.

KE, I thought, is a property of a system when viewed from a certain reference frame. Reference point shouldn't affect KE, should it? Or is reference point the same as reference frame here? (If reference point = reference frame, most reference frames are non-inertial, aren't they?)

4. Dec 2, 2014

### Staff: Mentor

You are welcome.

Changing the point of application of a force is not a coordinate transform. In a coordinate transform both the location of the force and also the location of the rod will change so that the force is applied to the same piece of material in all coordinate systems.

For simplicity let's fix the initial location of the rod so that the center of mass is at the origin, and the point of reference to also be at the origin. Now vary A and the final velocity of the center of mass will not vary, but both the KE and the angular momentum will.

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