Systems held together by gravity

[Wuratch]

Homework Statement

For any system of particles held together by mutual attraction
U(potential) = -2 U(Kinetic)

Suppose you add energy to such a system and wait for it to equilibrate, does the total kinetic energy increase or decrease? Explain

Homework Equations

The Attempt at a Solution

Apparently the kinetic energy decreases. How can you explain this?

 

[Dick]

Well, suppose you add energy to the earth-moon system. That moves the moon to a higher orbit. Does the kinetic energy decrease or increase?

 

[Wuratch]

Well, suppose you add energy to the earth-moon system. That moves the moon to a higher orbit. Does the kinetic energy decrease or increase?

It would increase and that's what I thought. But the book says that it is a system with a negative heat capacity. It's entropy vs. energy graph is concave-up. When you add energy, all the energy apparently goes into the potential energy and the particles in the system get farther apart and actually slow down. I don't fully understand this and don't know how to to give the explanation without just repeating what the book says

 

[Matterwave]

Actually the moon moving to a higher orbit means that it must slow down...

Note: mv^2/r=GMm/r^2 (M is mass of Earth, m is mass of moon)=> v=sqrt(GM/r) so that as r increases, v decreases.

Increasing the energy of a system in virial equilibrium goes like this:

E increases, so that T+U increases, but U=-2T, so that T-2T=-T increases. An increase in "negative T" means that T decreases and U increases. By U increases, we mean U becomes less negative.

For example, if E=-2J, T=2J, U=-4J, and I add 1J to this system, E now has to be -1J, and now I have a system of equations:

-1J=T+U, and U=-2T. Solving: -1J=T-2T=-T => T=1J, U=-2J. So you notice that U got less negative (went up) and T went down.

 

[Unown]

Hey, I tried to do this in this way:

I use $$V$$ for potential energy and $$T$$ for kinetic energy.

Given equation reads:

$$V=-2T$$

$$V+T=-T$$

Now you see that the left hand site is a total energy:

$$E_{tot}=-T$$

Differentiate the whole thing:

$$dE_{tot}=-dT$$

So if the total energy is increasing then $$dE_{tot}>0$$ and thus $$dT<0$$. This means that the kinetic energy is decreasing.

 

[Wuratch]

Thanks everyone. I realize my mistake was trying to explain it with just words