Linear and angular momentum

Question

If I push on an object not through the center of mass, we say that the translational effect on the entire object is the same. In other words, we can turn that off-center force into a force through the center of mass and a torque about it.

But if the force was originally through the center of mass, we wouldn't get a rotation.

I can see that in both cases, you are doing the same amount of work because the rotation caused by the force doesn't count towards work. But for the same force at different locations, it seems that one adds more energy to the system due to adding both angular momentum and linear momentum, while the other only adds linear momentum (of the same amount).

Is the linear momentum added to the system really the same regardless of where you push? It seems to me that a push at the center of mass should add more linear momentum due to not adding any angular momentum, but that's apparently not the case.

Answer 1

You are not doing the same amount of work in both cases. You can see this in two ways. 1) Total work is the change in kinetic energy, and the kinetic energy differs in the two cases (the one that rotates has an extra rotating kinetic energy term), 2) The displacement of the point where you apply the force is different, it will be larger in the rotational case, so the work will be different too.

Answer 2

I can see that in both cases, you are doing the same amount of work because the rotation caused by the force doesn't count towards work...

You can't compare them because you haven't given any information about either case that details the total work.

$W = F \times d$. You've specified a force, but not the distance it acts over.

but for the same force at different locations, it seems that one adds more energy to the system due to adding both angular momentum and linear momentum, while the other only adds linear momentum (of the same amount).

Again, you haven't specified a distance, only a force. For the off-center case, some of the energy will go into rotation and some into translation. But you have to give more information about how it is applied to do the comparison.