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Suppose a man sits on a freely rotatable light chair holding 2 dumbbells to his chest, he then extends his hands while maintaining the height of each dumbbells (so that gravitational of the system is constant). His angular speed is reduced while his moment of inertia increases.

Where did the lost energy go?

(I do know numerical proof that energy do loss, but I wanted to know how and where did the energy go, any proofs by any formulas of physics is appreciated)

Thank you in advance

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The reverse motion is easier to understand, assume he first has his arms stretched, and then pulls the weight to the center.
What happened? He did work. The man feels centrifugal force, and against this force (which is not constant, so you cannot just calculate $W=F\cdot s$) he does work.

Now the other direction of the motion is less intuitive. Because, unfortunately, humans are not good in gaining work by letting the system do work on them. Or in other words, in capturing potential energy like a spring - otherwise you could jump without having to do the whole work every jump.

But nevertheless, the work is done by the dumbbells on the arms of the human - the force on the hand is in the direction of motion, it could gain energy if it were a spring.

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  • $\begingroup$ So according to your answers, the human loses energy while the dumbbells are gaining but human loses more than the amount dumbbells are gaining due to work against centrifugal force? Another question, due to centrifugal force is fictitious, can it be said that it is centripetal force that does the work, but in opposite directions? $\endgroup$
    – Derpson
    May 12, 2016 at 8:20
  • $\begingroup$ No, that doesn't sound right. The work against the centrifugal force is the transmitted energy from the man to the dumbbell. That's not what one calls "loss". Some energy is lost, because the muscles have a low efficiency (they get hot, they can't just contract). $\endgroup$
    – Ilja
    May 12, 2016 at 8:26
  • $\begingroup$ And energy is lost in the other direction, when the weight does work on the arm. The human cannot make any use of this work. This also is lost - just as in the case of jumping you cannot retain the energy you had before landing and use it to jump up again $\endgroup$
    – Ilja
    May 12, 2016 at 8:29
  • $\begingroup$ And, again, the centripetal force is the force from the man on the dumbbell. So if motion is inwards, then the work (on the dumbbell) is positive, and if motion is outward, then contrary the dumbbell does work on the arm - which cannot make use of it $\endgroup$
    – Ilja
    May 12, 2016 at 8:32
  • $\begingroup$ So by your explanations, if the arms are now replaced by a pair of springs that can gain energy by being worked on, will the energy of the system remains constant? $\endgroup$
    – Derpson
    May 12, 2016 at 8:41

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