Classical mechanics: ball rolling in a hollow sphere

[blalien]

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

Homework Statement

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:
a: radius of ball
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

Homework Equations

I = 2/5ma^2

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?

 

[Shooting Star]

The Attempt at a Solution

First of all, I'm assuming that $$\omega$$=$$\theta$$'. It sounds intuitive, but I could be wrong there.

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

 

[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.