Solve Rotation Q: Period of Small Oscillations for Rolling Cylinder

[joker_900]

Homework Statement

A small uniform cylinder of radius a rolls without slipping on the inside of a large fixed cylinder of radius b (b>=a). Show that the period of small oscillations of the rolling cylinder is that of a simple pendulum of length 3(b - a)/2

Homework Equations

rotational KE = 0.5Iw^2

The Attempt at a Solution

OK I tried to do this with energy considerations. I called the angle between the vertical and the line from the centre of the big cyclinder to the position of the small one p, and so let the GPE be mg(1 - cosp) and expanded cosp to 2 terms. I added this to the KE of the centre of mass and the rotational energy 0.5Iw^2 of the cylinder.

I therefore need some sort of relationship between w, the angular velocity of the cylinder about it's centre, and the angle p. I thought perhaps adp/t=bw, as the point on the cylinder in contact with the big cylinder will have a tangential velocity described by both. But basically I'm stumped - please help!

 

[joker_900]

no one?