Collision with a ball spinning about its vertical axis

[Mmarzipan]

Homework Statement

A ball A is rotating on a table with an angular velocity ω about its vertical axis. An identical ball B collides with the ball A elastically. After collision the ball A starts sliding over the table. The coefficient of friction is µ. Find:

1) the angle α between the angular velocity vector of the ball A and the vertical for any moment of time before the ball starts rolling

2) the time instant when ball A starts rolling

3)what happens with the angular velocity of A after it starts to roll

Homework Equations

Momentum is convserved?

The Attempt at a Solution

For the first question, the answer seems to easy - the angular velocity axis should be always vertical, that's said in the text as well.
For the second one I tried many ways, conservation of energy and conservation of momentum, but there's no information about the initial and final velocity of the second ball so I'm not sure. Should I use integration?
Third should be easy too - the answer is that it stays the same (in ideal conditions).

 

[haruspex]

the angular velocity axis should be always vertical

I would not have thought so. There is a torque from friction normal to the angular momentum. Should it not precess?

 

[Mmarzipan]

Oh, of course! I have no idea how to calculate that tho. The friction force has a different direction at every point, so its torque is against the original spinning, right?
so
T = dL/dt = I*dw/dt = F*r*sina
F being frictional force and r the radius of the sphere. But I have neither of those given. W is omega, angular velocity.

 

[haruspex]

so its torque is against the original spinning, right?

Not sure what you mean by that.
At any time t, let the velocity of the sphere's centre be $\vec v$ and its angular velocity be $\vec\omega$.
What is the velocity at the point of contact with the ground? What frictional force does that produce? What are the consequences for the linear and angular velocity?
Introduce variables for mass and radius as necessary, but likely these will cancel out later.

 

[Mmarzipan]

I meant that it should slow down the spinning, or shouldn't it?
Okie, at the point of the ground angular velocity is practically zero because it's spinning around its vertical axis, shouldn't it be so?

, so linear velocity v = (r x ω) * r². But in the bottom point r and ω are antiparallel so linear velocity v is zero.

 

[haruspex]

I meant that it should slow down the spinning, or shouldn't it?
Okie, at the point of the ground angular velocity is practically zero because it's spinning around its vertical axis, shouldn't it be so?

, so linear velocity v = (r x ω) * r². But in the bottom point r and ω are antiparallel so linear velocity v is zero.

I defined vectors $\vec v$ and $\vec \omega$ as independent of each other. There is no equation relating them. This is not rolling contact.
Not sure how you are defining $\vec r$. If it is the vector from the centre of the sphere to some point on its surface, what is the velocity of that point?

 

[Mmarzipan]

Oh of course!
Linear velocity is v=r*ω for any rotation. I'd define r here as distance from the (vertical) axis. But in the point of contact, r is 0. So linear velocity is 0.

 

[haruspex]

Oh of course!
Linear velocity is v=r*ω for any rotation. I'd define r here as distance from the (vertical) axis. But in the point of contact, r is 0. So linear velocity is 0.

No, you are still missing the point, and that is not the right definition of r.

If the ball is rotating with angular velocity $\vec \omega$ then at the point of the ball $\vec r$ from the centre the velocity, relative to the centre of the ball, is $\vec r\times\vec\omega$.
But the ball's centre has linear velocity $\vec v$. So what is the velocity of the point relative to the ground?

 

[Mmarzipan]

I'm so trying! Where it touches the ground, r and w are antiparallel.
ω is about the vertical axis of the ball. Where the ball touches the ground, r also goes along the vertical axis.

 

[haruspex]

Where it touches the ground

My question was in regard to an arbitrary point on the surface of the ball. It is a simple question about relative velocities. I have given you the formula for the velocity of the point relative to the centre of the ball, and we know the velocity of the ball, so what is the velocity of the point?

ω is about the vertical axis of the ball

Only initially. We need to allow for arbitrary rotations later.