Skier on hemispherical slope problems

[fliptomato]

"Skier on hemispherical slope" problems

Greetings--I'm stuck on a mechanics question: Suppose you had a skier at the top of a hemispherical ski slope. S/He has some initial velocity. What is the maximum such velocity such that the skier maintains contact with the ski slope?
Similarly, how would I approach this if the cross section of the slope were not a semi-circle, but instead a cosine, or some other shape?

Thanks very much,
Flip

 

[Doc Al]

To maintain contact with the slope, the skier must be centripetally accelerated. That centripetal force is provided by a component of the skier's weight. At some point, that force will be insufficient to maintain contact. (Apply Newton's 2nd law.)

For an arbitrarily shaped slope, the same idea would apply but would be more difficult to calculate since the radius of curvature changes along the path.

 

[amcavoy]

For a circle, you also have:

$$a_r=\frac{v^2}{r}=mg\cos{\theta}$$