# Calculus-based energy problem

## Homework Statement

A uniform chain of mass M and length L hangs from a hook in the ceiling. The bottom link is now raised vertically and hung on the hook.
a. Determine the increase in gravitational potential energy of the chain by considering the change in position of the center of mass of the chain.
b. Write an equation for the upward external force F(y) required to lift the chain slowly as a function of the vertical distance y.
c. Find the work done on the chain by direct integration of F.

## Homework Equations

PE=W
change in PE= PEf-PEi

## The Attempt at a Solution

a. I found the change in PE from height L/2 to 3L/4 to be mgL/4
b. I understand that this part entails a mass density change, because the length is shortened while the mass stays constant, but I cannot figure out how to write an equation for the F based on this observation.
c. Integration of the force(answer in b.) will equal the answer in a. because PE=W, but I don't have an equation to work with.

Thanks to everyone who look the time to look at this problem!