What is the angle between a rolling ball and the surface it leaves?

[Ihatemiu]

Homework Statement

Situation: There is a big ball that never moves, and a small ball on it.
If we let the small ball roll down from the big ball, what is the angle that between the top of the big ball and the place that the small ball leaves the surface of the big ball?
Or do we need more information about situation?

Homework Equations

Maybe equation about circular motion?

The Attempt at a Solution

I tried to let R=0 to my calculations but I found out that the angle=0

 

[PeroK]

Did you make this question up yourself? Are you sure the small ball rolls and isn't slipping?

 

[Ihatemiu]

Did you make this question up yourself? Are you sure the small ball rolls and isn't slipping?

My teacher asked me today. What about the ball roll down without slipping?

 

[PeroK]

My teacher asked me today. What about the ball roll down without slipping?

That's a harder problem. What are your thoughts about what's happening and why the small ball eventually leaves the surface? Hint: circular motion is important here.

PS There was a problem on here not that long ago to prove that the small ball must slip before leaving the surface.

The simpler and solvable problem is to consider the small ball as a point sliding down the large ball.

 

[haruspex]

PS There was a problem on here not that long ago to prove that the small ball must slip before leaving the surface.

Yes, for any given coefficient of friction, sliding must occur before losing contact completely. On the other hand, the difference between those two angles can be made as small as you like by making the coefficient large enough. So by allowing an arbitrarily large coefficient, it is reasonable to treat it as though rolling contact is maintained.
With that simplification, is it much harder than the sliding point mass case?

 

[andrewkirk]

I suppose in the rolling case one needs to know the ratio of moment of inertia to mass for the small ball, in order to take proper account of its rotational energy, and that would depend on whether it was solid or hollow. Whereas in the non-rolling case one can reasonably take the small ball as a point mass, provided it's small enough.

 

[haruspex]

I suppose in the rolling case one needs to know the ratio of moment of inertia to mass for the small ball, in order to take proper account of its rotational energy, and that would depend on whether it was solid or hollow. Whereas in the non-rolling case one can reasonably take the small ball as a point mass, provided it's small enough.

True, but one could start with an arbitrary radius of inertia, k, and obtain a general solution.

 

[PeroK]

Yes, for any given coefficient of friction, sliding must occur before losing contact completely. On the other hand, the difference between those two angles can be made as small as you like by making the coefficient large enough. So by allowing an arbitrarily large coefficient, it is reasonable to treat it as though rolling contact is maintained.
With that simplification, is it much harder than the sliding point mass case?

It's a good point. If we simply assume that the ball keeps rolling until it loses contact, then it's a nice problem. It's a little bit harder than the point mass problem. And, it's simpler to assume that the small ball has a much smaller radius than the large ball.

My advice to the OP would be to do the point mass problem first, then consider a small rolling ball (and assume it doesn't slip).