Question

In the situation show find the maximum extension of the springs

Answer 1

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up vote

5

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accepted

The first method is giving the correct answer. In writing the work done by the force, you are assuming that the force FF itself is constant throughout the extension. However, this is not true. While extending the spring in a quasi-static way, the force FF must always match exactly the spring force at that time. This is needed so that at the end of the extension, the spring remains at rest. Once we understand this, we note that the force at extension xx is F(x)=kxFxkx. Then, the work done along the path is

∫xmax0F(x)dx=12kx2max

0xFxdx12kx2

The latter of course is precisely the potential energy of the spring. Thus, the "energy conservation" equation is trivial and does not yield any new information.