Motion in Nonlinear Differential Equations

[kgns]

Homework Statement

How do you derive the time-dependent velocity equation for motion along a curve, such as a skateboarder on a half pipe?

For the sake of abstraction, I ask myself the following:

A uniform sphere of mass m and radius r is set free from the top edge of a semicircle half pipe with radius R. If R > r, what is the time-dependent velocity equation v(t) for the sphere in terms of t, m, r, R and g?

(i) ignoring any effects of friction?
(ii) if the sphere is rotating?2. The attempt at a solution

If it were an inclined plane, we'd have no problem with
[itex]v(t)=gtsin(\alpha)[/itex]

Considering the halfpipe an infinitesimal sum of inclined planes we'd get
[itex]\int^{0}_{t}gsin\alpha(t)\partial\alpha[/itex]

However I've failed to derive [itex]\alpha[/itex] in terms of t.

How can I model such a problem in differential equations?

 

[haruspex]

I assume this is all in the plane at right angles to the trough's axis.
If you ignore friction then the object will slide, so its shape is somewhat irrelevant. The problem becomes a simple pendulum, but using the exact equation, not the approximation for small angles that makes it effectively SHM. You should be able to derive the ODE using the usual free body approach.
For a rolling sphere, I don't expect it to be much different. Should be equivalent to reduced gravity. Again, do try to obtain the equation.