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!

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ideasrule
Homework Helper
For b), you're on the wrong track. Density does stay constant because the total length of chain stays constant; you're not significantly stretching or compressing the chain.

Try drawing a free-body diagram on the part of the chain that's being raised--that is, the part between your hand and the lowest part of the U-shaped chain. Then write out Newton's second law, and you'll see what the force is.

I have drawn out the free-body diagram, but it still isn't clicking for me. How do you apply the calculus to this problem? Is mass constantly increasing, requiring more force as the chain is being raised?