Finding the potential energy

[rolypoly3000]

1. We have a cylinder of Mass M rotating about its axis. String is wound around it. Other end is connected to a spring. A mass m is attached to the spring.

The problem is to find the Lagrangian. My only issue is with finding the potential energy. The problem asks to use the distance of the mass m from center of axis of cylinder to be 'y' and length of spring to be 's'.

If I take downward positive, I can write,( with U=0 at axis of cylinder) I can write,

U=mgy + 1/2ks^2

But if I take downward negative, U= -mgy+1/2 k s^2

So using two coordinate systems will result in two different end answers for equations of motion.

Any thing wrong in what I am doing

 

[CompuChip]

This is just a sign convention and it should not matter in the final answer.
In this case I think that y and s are independent, right? So the equation of motion for y is just m g y'' = 0 in one convention and -m g y'' = 0 in the other, so it doesn't matter. When you are careful with the sign in imposing the boundary conditions, you should be fine.
Also, note that flipping the sign on y also flips the sign on s (so you get a different U, but that's ok - but differentiating you will lose the s^2).

So, to summarize, I don't think there is a problem. If you think there is, maybe you can post some details/