(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A thin rod of length 2l and linear mass density of [itex]\lambda[/itex] is constrained to move with its ends on a circle of radius a, where a>l. The circle is in the vertical plane (gravity is present). The contacts between the circle and rod are frictionless.

Part A: Write down the Lagrangian describing the motion of the rod.

Part B: Calculate the frequency of oscillation for small departures from equilibrium.

2. Relevant equations

The Lagrangian: L = [itex]\frac{1}{2}mv^2[/itex] + mgh. Where h is the height and is some function of the angle [itex] \theta.[/itex]

The Euler-Lagrange equation: [itex]\frac{d}{dt}[/itex][itex]\frac{\partial L}{\partial q}[/itex] = [itex]\frac{\partial L}{\partial \dot{q}}[/itex]

3. The attempt at a solution

The distance between the center of mass and the center of the circle (call itL) must remained fixed, the center of mass moves like a simple pendulum. Its Lagrangian is given by

[itex]L = \lambda l L^2\dot{\theta}^2 - 2\lambda gLcos\theta[/itex]and would have a period of [itex]\sqrt{\frac{L}{g}}[/itex].

I can't tell if I have to describe the motion at some point [itex]dl = \sqrt{dx^2 + dy^2} [/itex]from the center of mass (with coordinates [itex]X=Lsin \theta[/itex] [itex]Y =Lcos \theta[/itex]. I think it comes down to a problem with understanding the geometry of the problem. I also can't remember what ends up happening to the Lagrangian of an extended body.

Attached is a picture of the system.

Thank you for any help!

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# A rod constrained at it's ends

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