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If I start off in a crouched position, with my feet on the floor, and I extend my hips and knees so that I am now standing, my centre of mass has accelerated upwards, and work has clearly been done on me.

My understanding is that the centre of mass of a system cannot be accelerated unless there is a net external force acting upon it. When I extend my hips and knees, there are forces generated by my muscles, and there is a ground reaction force.

The ground reaction force is the only external force acting upon me, so it must be what is doing the work of accelerating my centre of mass upwards. Yet the ground reaction force can do no work.

Can someone help me resolve this apparent paradox?

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  • $\begingroup$ Related and there are many more physics.stackexchange.com/q/103473 $\endgroup$ – user176049 Nov 24 '17 at 2:49
  • $\begingroup$ Thanks. I had gone that one and a number of others, but didn't find any answers that addressed this particular question. I may have missed one though! $\endgroup$ – spacediver Nov 24 '17 at 3:14
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The experiment becomes easier to understand when we lift an approximation: that the Earth has infinite mass.

Now we have two balls (your torse and the Earth) connected by your legs: when they extend, the center of mass of the system stays in place, but the balls are further apart. Both balls moved, so the legs did work on both of them

Now, of course, since one of the balls mass is much larger than the other's, its displacement can usually be ignored.

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The forces generated by your muscles are doing the work

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  • $\begingroup$ Your feet don't move so $\vec{F}\cdot\vec{x}$ appears to be 0. Consider your hips/torso/and upward: your legs apply a force over a distance. Boom: work. $\endgroup$ – JEB Nov 24 '17 at 3:08
  • $\begingroup$ Yes, but the force that the muscles are applying are all internal. And all internal forces cancel out to zero net external force, right? $\endgroup$ – spacediver Nov 24 '17 at 3:10
  • $\begingroup$ Not if you change shape. $\endgroup$ – JMLCarter Nov 24 '17 at 3:17
  • $\begingroup$ Why does changing shape mean that internal forces don't cancel out? If I were in outer space, in a crouched position, and I extended my hips and knees, I am changing shape as I do so. Yet the net external force on me is zero, which is why my centre of mass does not experience an acceleration. $\endgroup$ – spacediver Nov 24 '17 at 4:46

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