Fandom

Scratchpad

J08M2

215,933pages on
this wiki
Add New Page
Discuss this page0 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

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}}

Also on Fandom

Random wikia