# Block on a Cylinder, using Lagrange's Equation

1. Feb 18, 2010

### Oijl

1. The problem statement, all variables and given/known data
A hard rubber cylinder of radius r is held fixed with its axis horizontal, and a wooden cube of mass m and side 2b is balanced on top of the cylinder, with its center vertically above the cylinder's axis and four of its sides parallel to the axis.

Assuming that b < r, use the Lagrangian approach to find the angular frequency of small oscillations about the top.

2. Relevant equations
T (kinetic energy) = (1/2)(mv^2 + I$$\dot{\theta}$$$$^{2}$$)
I (moment of inertia about the center of mass) = (2mb^2)/3
U (potential energy)= mg[(r + b)cos$$\theta$$ + r$$\theta$$sin$$\theta$$]

3. The attempt at a solution

Now, what would be nice would be to write the coordinates of the center of mass. I can differentiate that and get v, which I plug into T and then I have L = T - U and I can do the problem.

But how can I write down the coordinates of the CM? I never was very good at center of mass problems.