# I Can the centre of mass do work?

1. Nov 5, 2017

### albertrichardf

Hello,

Consider two equal masses moving away from each other at the same speed. The total momentum of the system is zero, so the total momentum of the system is zero. Therefore, the centre of mass has no speed.

The total energy in the system is $mv^2$, due to the kinetic energy of the two masses. The energy of the centre of mass must be stored in some other way, because the kinetic energy of the centre is zero. It would be the "internal energy" of the centre, just like an ideal gas has internal energy due to the kinetic energy of its molecules.

Suppose, now we create an ideal box, one for which all collisions with the inner walls are perfectly elastic. We can take the system we imagined, and place it in the box. The box is then sealed. Inside the box, the masses go back and forth, colliding with the walls of the box, and colliding with each other. However, it is arranged so that the total momentum of the masses at any instant is zero. The box then contains some energy $mv^2$.

My question is, can we extract that energy and do work with it without ever opening the box? Ideal gasses allow this energy to be detected by increasing the temperature of the box. By placing the box with something at a lower temperature we can extract work. Is there a macroscopic equivalent to that?

2. Nov 5, 2017

### Staff: Mentor

Sure. Let the collisions with the wall be inelastic. That is how heat is transferred too.

3. Nov 5, 2017

### albertrichardf

Thanks for answering. So there is no way of extracting energy, unless the collisions are inelastic?

4. Nov 5, 2017

### Staff: Mentor

Correct. That is the definition of an elastic collision.

Edit: note @jbriggs444 has a more clever solution below

Last edited: Nov 7, 2017
5. Nov 5, 2017

### albertrichardf

That makes sense: since the kinetic energy of a system with solely elastic collisions can't change, you shouldn't be able to extract any energy from it either. Thank for clearing it up.

6. Nov 5, 2017

### jbriggs444

Suppose that instead of a rigid box, we have one with a fixed bottom, tall fixed sides and a top that is free to slide up and down. i.e. A cylinder and a piston.

Balls inside the cylinder can collide with the top. These collisions are elastic -- total kinetic energy of top plus balls is unchanged. But the top is pushed upward as a result. One could harvest the energy of the moving top. The net effect would be to remove kinetic energy from the balls.

7. Nov 7, 2017

### albertrichardf

Thanks for the answer. It does make sense that such a system would behave like an ideal gas.