Does a force independently affect velocity and angular velocity, or is it split between the two?

Question

I'm implementing physics for a computer game, and came accross something that looks unintuitive to me.

Consider two bodies at rest:

Let's say we momentarily apply the same amount of force to them, but in different locations: at the center of mass for one body, and off-center for the other.

The second body will gain angular velocity, while the first one won't. Both will gain rightward velocity, in the same direction.

But will the second body gain less velocity than the first? Or the same velocity?

Naively I would expect less velocity, since some of the force was "expended" on giving it angular velocity, but I googled around and experimented in Algodoo, and it seems the velocity ends up the same.

Is that correct? Is there an intuitive explanation for that?

Answer 1

'Force' is the rate of change of linear momentum. When a rigid body rotates about its centre of mass, the linear momentum of one side of the body cancels the linear momentum of the other side moving in the opposite direction, so there is always a net zero momentum associated with the rotation. None of the linear momentum a force provides contributes anything to rotation.

Separately, a force has an associated 'torque', which is the rate of angular momentum about a point. The force through the centre of mass applies zero torque about the centre, and the off-centre force applies non-zero torque, so it's no surprise that one contributes angular momentum and the other doesn't.

Your intuition may be thinking about the force contributing to the energy of the body. There is linear kinetic energy associated with the motion of the centre of mass, and rotational kinetic energy associated with rotation about the centre of mass. The contribution of a force to energy is the force times the distance through which the force is applied. (It's called the 'work'.) When a force is applied through the centre of mass for a set time, the body moves only a small distance, as the entire body has to be accelerated. If the force is applied off-centre for the same time, the point of contact can be moved much further, because the acceleration is non-uniform. (To take an extreme case, imagine a dumbell with all the mass concentrated at the ends. Push on one end, and only half the mass gets accelerated, rotating about the other end which barely moves.)

Because the force is applied through a longer distance when off-centre, more energy is transferred by the force, and gets divided between linear and rotational kinetic energies.

Answer 2

The velocities will be the same. Inspired by @NulliusinVerba's dumbell example, I came up with an intuitive proof:

Let's say we split the ball in two, and momentarily apply the same force to the center of mass of one of the halves:

The upper half gets twice the velocity (compared to the first ball from the question, because it has half its mass), and no angular velocity.

Then we immediately attach the two halves together.

The velocity of the resulting body is the same as that of the first ball, because of the conservation of momentum.

And the angular velocity of the resulting body is obviously non-zero (which can be thought of in terms of conservation of angular momentum relative to the center of the ball).