First, there is no splitting between linear momentum and angular momentum per se. They have different units, you can't add them, and it makes no sense to say "this is 30% linear momentum and 70% angular momentum". But you can calculate how much energy is stored in linear motion and how much is stored in angular motion, and (at least at non-relativistic speeds), you can indeed add them. So you are on to something here.
But Newton's Laws don't lie. If you apply a force F, then a=m/F, and the fact that the object is spinning doesn't change the acceleration. Yet applying the force off-center does indeed seem to add more energy to the object: you're accelerating it just as much as if you applied the force on-center and you're also spinning it.
So how do you resolve this? A piece of general advice in physics (and math, and many other fields) is to state your assumptions and your questions precisely and unambiguously. The question is: if you apply an equal force to two objects of equal mass, and there are no other external forces involved, how can one accelerate faster? But just because the forces are equal doesn't mean that the work (energy applied) is the same. In fact:
W (work) = F (force) * d (distance)
Divide by a small unit of time:
P (power, which is work per unit time) = F * v (velocity, which is distance per unit time)
And that's the velocity of the point that receives the force. And if you look at the animation, you will see that the off-center force on the rotating box is applied to a (variable) spot on the box that is moving to the right. So the power needed to apply the force is larger, and more work is done.
(In fact, the excess velocity is ωr, so the excess power is Fωr = ωτ (angular velocity times torque), which is exactly the power needed to produce angular acceleration. So energy is conserved and all is well.)
I think in my head I was mixing up "force" and "power"; it's clear that with two cubes travelling linearly at the same velocity, the one that's also rapidly spinning has more energy.
That it can take varying amounts of energy to apply the same amount of force to the same object was the missing piece for me, since I was thinking of force as power.
It’s fairly easy to demonstrate this if you have a friend with a bicycle. Have someone on a bike stay still and hold the brakes, and press firmly on their back. It’s easy. Then have them bike at a slow jogging pace, run along with them, and try to apply the same force on their back. It will be hard work.
First, there is no splitting between linear momentum and angular momentum per se. They have different units, you can't add them, and it makes no sense to say "this is 30% linear momentum and 70% angular momentum". But you can calculate how much energy is stored in linear motion and how much is stored in angular motion, and (at least at non-relativistic speeds), you can indeed add them. So you are on to something here.
But Newton's Laws don't lie. If you apply a force F, then a=m/F, and the fact that the object is spinning doesn't change the acceleration. Yet applying the force off-center does indeed seem to add more energy to the object: you're accelerating it just as much as if you applied the force on-center and you're also spinning it.
So how do you resolve this? A piece of general advice in physics (and math, and many other fields) is to state your assumptions and your questions precisely and unambiguously. The question is: if you apply an equal force to two objects of equal mass, and there are no other external forces involved, how can one accelerate faster? But just because the forces are equal doesn't mean that the work (energy applied) is the same. In fact:
W (work) = F (force) * d (distance)
Divide by a small unit of time:
P (power, which is work per unit time) = F * v (velocity, which is distance per unit time)
And that's the velocity of the point that receives the force. And if you look at the animation, you will see that the off-center force on the rotating box is applied to a (variable) spot on the box that is moving to the right. So the power needed to apply the force is larger, and more work is done.
(In fact, the excess velocity is ωr, so the excess power is Fωr = ωτ (angular velocity times torque), which is exactly the power needed to produce angular acceleration. So energy is conserved and all is well.)