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Suppose I have a spring with one end attached to a wall and the other end attached to a piece of rope. Now imagine I pull on the rope and pull the spring outward. My hand expends energy and the spring gets potential energy.

Now if I let go of the rope, the potential energy gets converted to kinetic energy and the two convert to each other back and forth as the spring oscillates. Of course, over time the spring loses energy to friction as well.

But now imagine instead I move the rope slowly so that the spring slowly goes back into equilibrium. Here the potential energy of the spring was decreased but no kinetic energy was created (or at the very least there is an arbitrarily small amount of KE as I move my hand).

How can we explain where the energy went?

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    $\begingroup$ What research have you done to answer the question? Are you familiar with how our muscles apply force and dissipate energy through actomyosin contraction and release within the myofibrils? $\endgroup$ Jan 6, 2022 at 0:52

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The potential energy has gone into the work done by the spring resisting a force along a distance, $W=F*d$. If instead of your hand there were a heavy block sliding along a surface with friction, the force $F$ would be the friction. In the case of your hand pulling, it's harder to see, but you can think of your body as essentially an engine, and the force of pulling the spring as a load on the engine (forcing it to work a little harder, using a bit more energy per second than it was before it was pulling).

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