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Ok, before we get to my question, I will describe the problem. You have a diver who jumps off a bridge with a slight angular speed about his center of mass (figure 1). While in the air, he curls into a ball. Only gravity acts on the person while he is jumping, and since gravity acts on the center of mass of the person, gravity provides no external torque and hence the angular momentum of the person is conserved. He then uncurls and dives into the water with the exact same angular velocity he started with. During this whole process, imagine that no heat is generated (a.k.a human body is 100% efficient, no air resistance, etc.). Divers changing their angular speed

So this is where I get confused. Intuitively, I think that if you end up the same angular velocity, then its like the diver never curled up in the first place and instead had the same angular velocity the whole time. So, the change in kinetic energy should be (mass of person) x (height travelled) x (gravitational field) due to the work done by gravity. But, it takes energy to curl up and un-curl. The internal chemical energy must be converted to something, but it seems as if it's not converted to kinetic energy (which I assume is the only thing that it could be converted to (might be a wrong assumption)). So, where does the internal chemical energy go?

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You already have your answer:

But, it takes energy to curl up and un-curl.

Your muscles do work to move your body. That is where the "chemical energy" you are asking about goes.

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  • $\begingroup$ Yes, but when you arrive at the bottom, I think you have the same kinetic energy (I imagine that the scenario in which you curl and uncurl to the same angular speed is the same as if you never curled in the first place). But you have used internal chemical energy. So my question is where does this energy go if it doesn't go into your kinetic energy? $\endgroup$
    – Imajinary
    May 5, 2020 at 18:45
  • $\begingroup$ @Imajinary Energy has to be used in order for muscles to work. Even in the case of no energy change of "external objects", like holding an object at a constant height. The molecular motors in your muscles use energy to keep the muscles contracted $\endgroup$
    – BioPhysicist
    May 5, 2020 at 18:48
  • $\begingroup$ If you were in an idealized world (everything is 100% efficient, no energy lost to heat, it is my understanding that if you held an object at a constant height that you would not have to expend energy as you are not losing heat and not doing work. The reason why you do feel tired in the real world is because you are not 100% efficient. Even if this isn't the case, when you un-curl there are no muscles that are contracted (all of them are relaxed). $\endgroup$
    – Imajinary
    May 5, 2020 at 18:51
  • $\begingroup$ @Imajinary "efficient" does not mean "no energy expenditure". Efficiency for your muscles would be comparing the chemical energy they use to the energy you can output to do things "externally". But even if you were 100% efficient you still have to use that energy. $\endgroup$
    – BioPhysicist
    May 5, 2020 at 18:57
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    $\begingroup$ @Imajinary, Your body heats up (gains thermal energy) when you exercise your muscles without doing work on anything. $\endgroup$
    – The Photon
    May 5, 2020 at 19:14

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