Finding Potential Energy for a Chain on Pulley System

[starryskiesx]

Hi there,
I'm having some problems trying to write down the Lagrangian of the following system:

A uniform flexible chain of mass M and length L is hung under gravity on a frictional pulley of radius a and moment of inertia I whose axle is fixed at a point above the ground. Write down the Lagrangian of the system using the generalised coordinate l denoting the displacement below the axle of one end of the chain. You may assume that L is sufficiently long enough so that some part of the chain hangs freely from both sides of the pulley.

So I've tried to make a constraint equation: I thought L = pi*a + l + l' where l and l' are the displacements below the axle of each end of the chain.

I'm also thinking the kinetic energy should be easy, we have a rotational kinetic energy from the pulley and also a translational kinetic energy from the motion of the chain.

The real problem is trying to write down what the potential energy of the system is, I'm getting confused about where the centre of mass will be.

Any suggestions? Thank you!

 

[kuruman]

The real problem is trying to write down what the potential energy of the system is, I'm getting confused about where the centre of mass will be.

You can break down the mass of the chain into three pieces: hanging piece 1 of length $l'$, hanging piece 2 of length $l$ and the semicircular piece 3 of length $\pi a$. It should be easy to find the CM of each piece separately. Then the CM of the whole chain will be given by the usual equation $$\vec R=\frac{m_1\vec r_1+m_2\vec r_2+m_3\vec r_3}{m_1+m_2+m_3}.$$