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When I lift my hands above my head, my center of mass moves upwards. What force causes my center of mass to move upwards?

We know that $$\vec{F}_{\text{net external over all particles}} = M\ddot{\vec{R}}_{cm} \tag{1}$$ where $M$ is the total mass and $\vec{R}_{cm}$ is the center of mass of the system. This is the equation of motion for the center of mass. It says that only external forces determine the trajectory of a center of mass.

Thoughts :

When I lift my arms above my head, this force could be an external force. If so, my question disappears. However, a new question arises which is: where is this external force coming from? It's source/location? (One would think the food that we eat... However I'm still confused how an external force can reside within our bodies/midochondria).

Now equation $(1)$ assumes that all internal forces are equal and opposite. If we assume this is not the case, equation $(1)$ becomes

$$ \sum_{\alpha}\sum_{\beta\neq\alpha}\vec{F}_{\alpha\beta} + \sum_{n} \vec{F}_{\text{n}}\;^{\text{external}} = M \ddot{\vec{R}}_{cm}$$

where the double sum takes care of internal forces and runs over all particles (the force on particle $\alpha$ due to all other $\beta$). So maybe it's the internal forces which allow my center of mass to shift upwards when I raise my arms. However, I'm still confused what this would even mean.

With play-doh if you deform the shape, it's easy to see why the center of mass would move. You are applying an external force to the play-doh. The human body is much more complicated, but I was wondering if I'm missing something in my above logic. (there is also a subtlety between a discrete and continuous mass distribution - but I think the equations still apply - or maybe they dont)

sammy gerbil
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DWade64
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2 Answers2

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When you raise your hands, your center of mass moves upwards because of the reaction of the support (floor or ground). If you raised your hands in space, your center would not move (if there are no gravity forces).

akhmeteli
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  • If you jump off the ground and suddenly raise your arms, would this shift the center of mass? Or if a skydiver does something with his arms, would that shift his center of mass? In both cases, would you say that the surrounding fluid (air) provides the stresses/forces needed to shift the center of mass? – DWade64 Feb 04 '18 at 17:20
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    @DWade64 : In the first case, the air resistance will be negligible, so the center of mass will not move significantly compared to the case when one jumps and does not raise their arms "in flight", in the second case, the air resistance will be comparable to the gravity force, so this case may require more detailed analysis. – akhmeteli Feb 04 '18 at 17:47
  • Ok this is my last question. If I laid horizontally on the floor and slide my arms above my head, my center of mass would shift. Would this be attributed to the floor as well? – DWade64 Feb 04 '18 at 17:56
  • @DWade64 : If you mean you move your arms in a horizontal direction, then yes, the movement of the center of mass will be attributable to the floor, but not because of the normal reaction of the support, but because of friction. The center of mass will not move if your arms move in a horizontal direction when you lie on a very smooth floor (no friction). – akhmeteli Feb 04 '18 at 18:01
  • Possible duplicate of Why normal doesn't do any work in this case? or Is There Any Work Done By Normal Reaction Force even if point of application of force is not getting displaced? – sammy gerbil Feb 04 '18 at 18:27
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Let us assume that you are floating freely in space. When you raise your hands above the head, you center of mass moves up towards your head because you change the shape of your body so that the center of mass moves relative to other parts of your body. Think of your play-doh example.This is accomplished by your internal muscular forces. The movement of the center of mass is, of course, only relative to some fixed point of your body, like e.g your nose tip. Viewed from an outside coordinate system, your center of mass doesn't change at all. When your hands move up, your body moves down so that the center of mass doesn't move. When you stand on earth and jump up throwing your hands up, your center of mass will, of course, move relative to earth because there is a force between your body and earth.

freecharly
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