the equipartition theorem: Using the equipartition theorem, and taking room temperature to be $300K$ :

$($a$)$ Calculate the root mean square displacement from equilibrium, at room temperature, of

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a $1$ gram mass attached to a spring with spring$-$constant $K=1\mathrm{.}0\frac{N}{m}.$ Consider only the displacement in the direction parallel to the spring.

$($b$)$ Calculate the root mean square velocity of the mass in part $($a$).$ You can either con$-$

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sider only the component of motion parallel to the spring, or in all three perpendicular directions. $($Perhaps better to consider all three, but either will get full marks.$)$

$($c$)$ Calculate the root mean squared angular velocity, $\sqrt[\phantom{2}]{{}^2},$ of an $O_2$ molecule at room

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temperature, for rotation about an axis perpendicular to the oxgyen$-$oxygen bond. $($One oxygen atom has a mass of approximately $16$ atomic units, where an atomic unit is $1\mathrm{.}66{10}^{-27}kg.$ The $O-O$ bond$-$length in an $O_2$ molecule is $121pm.)$ In calculating the moment of inertia, treat the oxygen atoms as point masses. $($This is a good approximation, since $99\mathrm{.}9\%$ of the mass is in the nucleus.$)$

$($d$)$ At low temperature, it is possible to condense an atomically thin layer of helium on the

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surface of an atomically flat graphite sheet" In this state, if you are careful, you can get a two$-$dimensional gas: the helium can move freely parallel to the surface of the graphite sheet, but it is trapped in a roughly harmonic potential well perpendicular to the sheet.

If we apply the equipartition theorem, what is the average thermal energy per helium atom in such a film?