We can crudely model a nitrogen molecule as a pair of small balls, each with the mass of a nitrogen atom, $2.3 \times 10^{-26} \mathrm{~kg}$, attached by a rigid massless rod with length $d=2 r=188 \mathrm{pm}$.
(a) What is the moment of inertia of this molecule with respect to an axis passing through the midpoint and perpendicular to the molecular axis? (b) Consider air at 1 atm pressure and $20.0^{\circ} \mathrm{C}$ temperature. Suppose two nitrogen molecules with rms speeds $v_{\mathrm{i}}$ have an elastic collision such that only one nitrogen atom in one molecule collides with one nitrogen atom in the other molecule in the manner depicted in Fig. P18.81. Write equations for the conservation of energy and the conservation of angular momentum, in terms of $r, v_{i}, v_{f},$ and $\omega,$ where $v_{f}$ and $\omega$ are, respectively, the center-of-mass speed and the angular speed of either molecule after the collision. (c) Solve these equations for $v_{\mathrm{f}}$ and $\omega$ in terms of $v_{\mathrm{i}}$ and $r .$ Note that $\omega \neq 0$ after the collision. (d) Using $v_{\mathrm{i}}=v_{\mathrm{rms}}$ for the conditions specified above, what is the frequency of rotation?