How Does Angular Velocity Relate to Harmonic Motion in a Rolling Ball?

[cosmic_tears]

Hi.
Ok, I've given this question already soooooo much time and I simply cannot solve it.

Homework Statement

There's a circular surface that's holding still (like a bowl), with radius R, and a ball, with radius r, on it.
The ball is rolling without sliding.
The mass of the ball and it's moment of intertia are given.
1. First question - express the relation between W, which is the angular velocity of the ball, and "d(theta)/dt", where "theta" is the angle formed in any time between the "main axis of the bowl" and the radius streaching to the ball.
2. Second question - find the relation between W and theta(not theta dot as before), given that the ball starts it's movement at hight "h" above the surface.
3. Third and last - given that the ball is oscillating in small values of theta, what is the period time of the harmonic movement? (they give a hint: differentiate (with t) the function we found in the last questions - W(theta), find a "movement equation", and compare it to the "harmonic oscillation" classic equation.

!

The Attempt at a Solution

1. I Think I did that - w(t)* r = (theta dot) * R.
2.Ok, here I used mechanical energy cons. and after some effort found a pretty complex relation between W and theta. I won't specify it here but it has square root and all :)
3. Here's the real trouble:
From their hint I think I need to find d(W(t))/dt ? So I can do that using the "chain law" (I'm not sure if that's the name) and take the derivitive of t like this:
d(W(t))/dt = d(W(theta))/d(theta) * d(theta)/d(t)
After doing that, using the relations I got in the previous questions, I get a non-linear, second-order diffrential equation. Not solvable of course.

The excercise is a pretty classic one, just a plate with a ball rolling on it - but I still find it very complicated! Maybe I'm over-complicating things?

I'm desprate for help :-\

Thank you very much for reading.

 

[learningphysics]

Since they are referring to small $$\theta$$, approximate $$sin\theta\approx\theta$$ and $$cos\theta\approx1$$.

Then you get a linear diff. equation. It should the equation for a simple harmonic oscillator.

 

[cosmic_tears]

I know that. I've tried that. Still not a linear diff. equation.

If I must, I'll post my calculations...
However, it's so hard to write them in here...

Thanks anyway.

 

[learningphysics]

I'm getting a linear differential equation... there's a major cancellation (simplification) that happens when you substitute in $$\frac{d\theta}{dt}$$ into your equation for $$\frac{d^2\theta}{dt^2}$$

You can get $$\frac{d\theta}{dt}$$ in terms of $$\theta$$ using your 2 equations for $$\omega$$... the one you get in the first part, and the one you get in the conservation of energy part.