Can a Ball Roll in a Horizontal Circle Inside a Bowl?

[Nick886]

circular motion problem??

hey all, i have to talke app maths for a semester and it was all fine until we got to modeling, now haven't a clue? can anyone help with this please?
a ball rolls under gravity inside a smooth bowl with circular cross section and inner radius"a". show that the ball can roll in a horizontal circle at a height "h" above the bottom of the bowl provided
v^2 = gh(2a-h/a-h). thanks

 

[gneill]

What do you know about circular motion? Can you calculate centripetal acceleration?

Have you drawn a diagram of the setup, and a free-body diagram for the forces acting?

 

[Nick886]

ye i have done that, well tried! there is no friction because its a smooth bowl. so is it just gravity or am i missing something obvious?? Probably! i got as far as the cent acc and have $$\omega$$^2 = g(tan$$\vartheta$$ / a)

 

[gneill]

Hint: Consider for a moment, in an appropriate frame of reference where such things are not frowned upon, the sum of the centrifugal acceleration and the gravitational acceleration that the ball feels. The bowl's reaction is equal and opposite, so draw in the reaction acceleration. Here's the tricky bit. If this vector is not aligned with the surface normal of the bowl, that is, aligned with the radius vector of the bowl (curvature radius a) at the point of contact, then there will be a net acceleration away from the point of contact, either upward or downward, tending to shift the height of the ball.

If this acceleration vector must be aligned with the radius vector for stability of the rotation height, then a simple ratio follows.