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Is the spring in a clock a torsion spring? As opposed to looking like a clothespin spring, its windings are all in the same plane, does it still work the same way?

Related Question: Energy stored in a clothespin spring (non linear spring)

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There are probably lots of different types of clock spring, but I'd guess you're thinking of something like this:

Clock Spring

This is not a torsion spring. The spring is basically a long strip of metal and the energy is stored by bending the strip of metal not twisting it. Like the example in your previous question, the reason for coiling the metal strip is to fit a long length of strip into a small space. You need a long length of strip so the deformation of the strip doesn't exceed the elastic limit.

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