Solution - Winding Spring

Assume the spring consists of N small springs, where N>>1. Hence, relaxed length of the small spring is .
The large spring equals to the small springs in series, so

From the equation above, we can conclude that multiply between spring constant of a part and its relaxed length is constant. And we can write

differentiate it, and we get

The total length of unwinding spring is L. If the cylinder rotates by small angle , a small part of the spring with total length will wind the cylinder, while the other part deformed to length L.
Relaxed length of the small part is,

Because the small part will wind the spring, the change of relaxed length of unwinding spring is,

Then, from equation (1) and (3), we can get

Integrate the equation, we will get

When the unwinding spring have relaxed length l, it will give force to the cylinder,

and give torque,

Thus, the change of energy when it rotates by small angle is

Substitute kl from equation (1),

Substitute k from equation (4),

Integrate two sides of equation,

And now, substitute and , and we get

Substitute for uniform solid cylinder.

That's my solution.
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