J08M2

J08M2 is a short name for the second problem in the Classical Mechanics section of the January 2008 Princeton University Prelims. The problem statement can be found in the problems list. Here is the solution.

Let $$\theta$$ be the angle between the vertical and the line that joins the center of the cylinder and the center of the sphere. Let $$\phi$$ be the angle by which the sphere rotates about its axis as it moves inside the cylinder. Because the sphere rolls without slipping, we must have:

$$R\theta=\frac{R}{2}\phi$$

We can now write the Lagrangian:

$$\mathcal{L}=\frac{1}{2}m\left(\frac{R}{2}\right)^2\dot{\theta}^2+\frac{1}{2}I(2\dot{\theta})^2-mg\left(R-\frac{R}{2}cos\theta\right)=\frac{13}{40}mR^2\dot{\theta}^2-mgR\left(1-\frac{1}{2}cos\theta\right)$$

$$\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=\frac{13}{20}mR^2\dot{\theta}$$

$$\frac{\partial\mathcal{L}}{\partial\theta}=-mgR\frac{1}{2}sin\theta$$

$$\frac{13}{10}R\ddot{\theta} \approx - g \theta$$

$$\omega=\sqrt{\frac{10g}{13R}}$$