Idea of Rolling, Rotation, and Transition

[Biloon]

I'm very confuse about the idea of applying Newton's law of forces and torques on rolling body. The book shows that you can separate the situation in rolling into pure transition and pure rotation. However, what I don't get is, for the calculation of Newton's 2nd law of forces on pure transition, they calculate all forces, not just the one through center of mass... If what I remembered, the forces that are not through center of mass should create both rotational and transitional kinetic energy, right? In case of rolling, the 2nd law simply applied as sigma F = ma, where F is applied anywhere on the body.

From that equation, I got myself into some contradiction. In case 1, If i have a Force F applied on the center of mass, then it accelerates with F = ma. In case 2, if the force is not directed on the center of mass, and if I follow the book, the book simply use Newton 2nd law of forces and torques... so my accelerations remain the same but with some extra torque. Therefore, for the same amount of force and over certain period of time, both cases will not have the same kinetic energy...

From what I believe, the force that is not on the center of mass will be distribute into rotational and transitional kinetic energy. However, the book simply applied full magnitude of forces not through center of mass for Newton 2nd law in rolling motion; that is really disturbing for me.

 

[Doc Al]

Don't confuse the application of Newton's 2nd law (ΣF = ma) with the work done by the applied force. As far as Newton's 2nd law goes, the point of application of the force doesn't matter when calculating the acceleration of the center of mass. But it certainly does when calculating the work done by the force.

If you apply the force off-center (so it exerts a torque about the center of mass) then the point of application of the force will move through a greater distance, which requires more work done and thus ends up giving the object more overall kinetic energy. The translational KE of the center of mass doesn't change but the rotational KE does--the 'extra' work goes into rotational KE.