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I have a doubt about the work done by elastic force.

The general formula of the work is: $W_{Fe} = \frac{1}{2}k(x_0^2-x^2)$; if we take that the state in which the spring is at rest, we have: $W_{Fe} = -\frac{1}{2}kx^2$ for both cases where the spring is compressed and stretched. But, isn't this contrary to the fact that work is a conservative force? since that $W_{Fe}+W'_{Fe}=$ $-\frac{1}{2}kx^2-\frac{1}{2}kx^2 \not = 0$

($W$ represents the case in which the spring passes from the compression position to the stretched position, and W 'the other way round)

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The minus sign in Hooke's Law tells you that the direction of the restoring force is opposite to the direction of the force that must be applied when the spring is stretched or compressed. A new sign convention must be used when calculating work done on a spring that goes from stretching to compression, because forces must be applied in opposite directions to do that work.

Also note that when the spring is stretched and you slowly lower the force on the spring to let it go back to the equilibrium position before you apply compression to it, the spring is doing negative work to arrive at that equilibrium position, assuming that the direction of the stretch is the positive direction. Thus, when you stretch the spring and then let it relax back to its equilibrium position, the net work done is equal to zero. Obviously, a similar argument applies when you are compressing the spring, where the work of compression is negative and the spring does positive work to get back to the equilibrium position, again assuming that the direction of spring stretching is positive.

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You may need to carefully define your symbols. The work you do in stretching or compressing a spring a distance x from equilibrium is positive. The work done by the spring in that situation is negative, but you are giving it positive potential energy. If you ease it back to equilibrium, it will be doing positive work and you negative. If you release it from a position of positive potential energy, energy will conserved as it goes from potential to kinetic, and then again from kinetic to potential after it passes through equilibrium.

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