# Angular momentum and collisions/pure rolling

Consider a general problem of a rigid sphere having only rotational motion placed on a rough surface where it starts rolling after some time. And final velocity is asked. Friction is just enough to provide rolling motion. All the books and class notes I've referred to say that to conserve angular momentum, external torque should be zero. External torque of friction is zero about the bottom most point. We can apply conservation of angular momentum about this point. So what I did was find moment of inertia about the bottom most point and give it only angular velocity. But the books just found it about centre of mass($$I\omega$$) + (mass * velocity * radius). I know both values come out to be the same, but from what I understand, we cannot apply it about centre of mass as external torque about centre of mass is not 0. Is my reasoning correct?

Now after starting rolling, it collides with a wall inelastically so that linear velocity becomes say v/2, and it returns to the direction it came from. But angular velocity just after contact remains the same in magnitude and direction as it was before collision. I was able to reason out the direction of angular and linear acceleration by taking friction opposing angular velocity but did not understand why angular velocity did not change direction or magnitude?

## 1 Answer

we cannot apply it about centre of mass as external torque about centre of mass is not 0. Is my reasoning correct?

Once the ball enters pure rolling friction force stops acting hence you can apply conservation of angular momentum but before that you can't.

Now after starting rolling, it collides with a wall inelastically so that linear velocity becomes say v/2

Is wall moving along with ball?

Anyways.

I was able to reason out the direction of angular and linear acceleration by taking friction opposing angular velocity but did not understand why angular velocity did not change direction or magnitude?

Just after the collision, yes angular velocity remains same since the normal reaction of wall on the ball can't produce any torque since it's pass through centre of mass.

But it will change after some time as friction will produce torque in opposite direction.

I am assuming wall to be frictionless.

• Friction with the wall during the collision could produce an impulsive torque, and change the angular momentum. – R.W. Bird Jul 24 '20 at 14:19
• @R.W.Birdr yes it can . My mistake.I edited my answer. – Danny LeBeau Jul 24 '20 at 14:22