Classical mechanics: ball rolling in a hollow sphere

1. Feb 22, 2008

blalien

[SOLVED] Classical mechanics: ball rolling in a hollow sphere

1. The problem statement, all variables and given/known data
This problem is from Gregory:

A uniform ball of radius a and centre G can roll without slipping on the inside surface of a fixed hollow sphere of (inner) radius b and centre O. The ball undergoes planar motion in a vertical plane through O. Find the energy conservation equation for the ball in terms of the variable $$\theta$$, the angle between the line OG and the downward vertical. Deduce the period of small oscillations of the ball about the equilibrium position.

So in summary, we have:
m: mass of ball
$$\theta$$: angle of the ball's position, relative to the vertical line connecting the center and bottom of the hollow sphere
I: moment of inertia of ball
$$\omega$$: rotational velocity of ball
T: kinetic energy of ball
V: potential energy of ball (V=0 at height $$\theta$$=$$\pi$$/2, the center of the sphere)
E: total energy of ball
g: acceleration due to gravity

2. Relevant equations
I = 2/5ma^2

3. The attempt at a solution
First of all, I'm assuming that $$\omega$$=$$\theta$$'. It sounds intuitive, but I could be wrong there.

I'm given, as a solution, that the period of small oscillation (that is, sin($$\theta$$)=$$\theta$$) is 2$$\pi$$(7(b-a)/5g)^(1/2), which I'm not getting in my results. I have a very strong hunch that my mistake comes from bad energy equations. So, would you mind taking a look of these?

T = 1/2mv^2 + 1/2I$$\omega$$^2
v = $$\omega$$*(b-a)
So T = 1/2m($$\omega$$*(b-a))^2 + 1/2(2/5ma^2)$$\omega$$^2
T = m$$\omega$$^2/10(7a^2-10ab+5b^2)
V = -(b-a)mgcos($$\theta$$)

So E = T + V = that stuff
Am I correct here?

2. Feb 22, 2008

Shooting Star

That is wrong. Draw a simple diagram to figure it out.

3. Feb 22, 2008

blalien

Hah, it's always the little mistakes in the beginning that steal away an hour of my life.

That fixed everything. Thank you so much for catching that.