Identifying the Potential Energy of a Spring Pulley System

[whitejac]

Homework Statement

Homework Equations

U = 1/2 kx²

The Attempt at a Solution

So, I took dynamics off campus and needless to say, we didn't really do very much deriving so that aspect of physics really bogs me down and I'm trying to learn it now in another class. So if anyone wants to also add their approach or method to their answers, that would be greatly appreciated because I don't just want to learn it for specific cases.

To identify the potential energy, I imagine the pulleys without the mass. The right pulley would pull vertically. The wire or spring connecting the pulleys to each other would tracking around the circumference of the pulley. So the length it would move as it rotated would be
x = 2rθ

The left pulley is fixed, so it cannot move. The mass is not present in my understanding of the potential energy situation. This means that:

U = 1/2kx² = 1/2k(2rθ)²

Is this correct?

 

[haruspex]

the length it would move as it rotated would be
x = 2rθ

Yes, depending on exactly how you are defining x. Don't forget that in addition to the rotation of the right pulley the rightmost spring may stretch. I suggest you create an additional unknown for that.

U = 1/2kx²= 1/2k(2rθ)²

Which spring is that for? Don't you need the PE of each spring?