# Finding Potential Energy for a Chain on Pulley System

• starryskiesx
In summary, the conversation discusses the problem of writing down the Lagrangian for a system consisting of a uniform flexible chain hung under gravity on a frictional pulley. The potential energy of the system is the main concern, with the challenge of determining the center of mass adding to the difficulty. The solution involves breaking down the mass of the chain into three pieces and finding the center of mass using an equation.
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!

starryskiesx said:
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 $$\vec R=\frac{m_1\vec r_1+m_2\vec r_2+m_3\vec r_3}{m_1+m_2+m_3}.$$

## What is the "Lagrangian of Chain on Pulley"?

The Lagrangian of Chain on Pulley is a mathematical expression used to describe the dynamics of a chain that is moving over a pulley. It takes into account the kinetic and potential energies of the system, as well as any external forces acting on the chain.

## How is the Lagrangian of Chain on Pulley derived?

The Lagrangian of Chain on Pulley is derived using the principles of classical mechanics, specifically the Lagrangian formalism. This involves using the Lagrangian function, which is defined as the difference between the kinetic and potential energies of the system, to derive the equations of motion for the chain and pulley system.

## What are the main variables used in the Lagrangian of Chain on Pulley?

The main variables used in the Lagrangian of Chain on Pulley are the position and velocity of the chain and pulley, as well as any external forces acting on the system. These variables are used to calculate the kinetic and potential energies of the system, which are then used to derive the equations of motion.

## What is the significance of the Lagrangian of Chain on Pulley?

The Lagrangian of Chain on Pulley is significant because it allows for the accurate prediction of the motion and behavior of a chain and pulley system. It is a fundamental tool in the field of classical mechanics and is used in various engineering and physics applications.

## Are there any limitations to the use of the Lagrangian of Chain on Pulley?

Like any mathematical model, the Lagrangian of Chain on Pulley has its limitations. It may not accurately describe the behavior of a system that is subject to large deformations or high speeds. Additionally, it assumes that the chain and pulley are ideal and do not experience any friction or other external forces.

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