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A man whose mass is $m$ jumps vertically into the air from a sitting position in which his center of mass is at a height $h_1$ from the ground. When his feet are just about to leave the ground, his center of mass is $h_2$ from the ground and finally his center of mass rises to $h_3$ when he is at the top of the jump. What is the average upward force exerted by the ground on him?

I know the foot, i.e. the point of contact, never moves. So the floor is not doing any work on the man. But the man is gaining energy so something is definitely working on him. I have a faint idea that it has got something to do with muscles but I am not very sure about it.

Vincent Thacker
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Satya
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3 Answers3

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I know the foot i.e. the point of contact never moves so the floor is not doing any work on the man.

Yes, that is correct.

But the man is gaining energy so something is definitely working on it.

Actually, the man is not gaining energy. The man is converting chemical potential energy into kinetic energy, and the sum of the chemical potential energy plus the kinetic energy is constant*. So no energy is being transferred to the man and thus no work is being done on the man, it is purely an internal conversion of energy from one form to another.

*Technically it should be the chemical potential energy + kinetic energy + thermal energy + gravitational potential energy that is constant.

Dale
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The man is not a rigid body. When the man extends his legs, his feet remain stationary, but his head and shoulders start moving upward. To make matters more complicated, his center-of-mass moves upward, but not as fast as his head and shoulders move.

A lot of the time, when we're doing physics, we look for ways to simplify our description of a complicated system. Center of mass is one such simplification. When we want to know, for example, the potential energy associated with the man's height above the Earth, we can use his center-of-mass instead of trying to account separately for the different motions of his head and his arms and his legs, etc., and we still get an answer that "works out."

If we treat the Earth as an immovable, rigid object (by far, the easiest way to understand the problem) then it probably makes sense to say that the man's legs are doing work on his own center of mass.

Solomon Slow
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Considering the center of mass (CM) as a point particle, the force from the floor does work on the CM. However, considering the entire man (not a rigid body) no work is done. To clear up this confusion, some employ the term "pseudowork" for the work on the CM, and reserve the term "work" for the broader concept as defined in thermodynamics. This is discussed in detail in my answer to If I jump off the ground is the net reaction force doing work on me?. An article by Arons discuses this, available online. [Development of energy concepts in introductory physics courses: American Journal of Physics: Vol 67, No 12 , A.B Arons] This reference explicitly address the jumping man case.

None of my 50-year-old physics texts dealing with classical mechanics addressed any of this, and I was confused about it for a long time until I found discussions such as in the reference above that cleared it up for me.

Hope this helps.

John Darby
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