Using Equipartition theory to solve the root mean square of a angle.

[Jameskd]

Homework Statement

Any tips on how to solve this?

Relevant Equations

1/2mv^2 + 1/2kx^2
3/2nRT = E

For the first question, i believe that mechanical energy is conserved hence we can derive the total energy i think. In regards to the second question, I'm assuming its at room temperature, so helium is monotonic therefore it has 3 degrees of freedom, therefore its internal energy is 3/2KT. I am unsure how this relates to the question.

Finally for part c, this is a shot in the dark, but the average kinetic energy of helium is 3/2NKT , so if equate 3/2NKT = 1/2(torsion constant)(torsion angle)^2 and solve for it i end up getting something that doesn't make sense.

 

[haruspex]

In regards to the second question, I'm assuming its at room temperature, so helium is monotonic therefore it has 3 degrees of freedom, therefore its internal energy is 3/2KT. I am unsure how this relates to the question.

Is it a matter of treating the mirror as just another particle in the mix?

 

[TSny]

Hello, jameskd.

For part (a) I think they want you to answer the following:
What property of a mass on a spring corresponds to $I$ in the energy equation for the mirror?
Similarly for $\omega$ , $\mu$, and $\theta$?

For part (b) the fact that the gas in this problem happens to be helium is not relevant. What's important is the number of degrees of freedom of the mirror system, not the number of degrees of freedom of a helium atom. You'll need a clear statement of the equipartition of energy theorem for this part.

Finally for part c, this is a shot in the dark, but the average kinetic energy of helium is 3/2NKT , so if equate 3/2NKT = 1/2(torsion constant)(torsion angle)^2 and solve for it i end up getting something that doesn't make sense.

Here it looks like you are equating the total kinetic energy of all of the helium atoms to the potential energy of the mirror system. That would mean that if you doubled the number (N) of helium atoms, the potential energy of the mirror would double. But there is no way for the mirror to "know" how many gas molecules are in its environment. The mirror is going to behave the same whether it's in a small flask of helium at temperature T, or in an large room filled with air at temperature T.

The average potential energy $\frac 1 2 \mu \langle \theta^2 \rangle$ of the mirror can be obtained from the equipartition of energy theorem.