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Let's consider gas of particles which move with constant velocity $V$. In such case we will derive speed of sound. $c_s^2=\frac{\partial P}{\partial \rho}$ where $P$-is a pressure and $\rho$ is a density of gases. Set p is a momentum, then the momentum which gas give the wall from small time $\Delta t$ is $\Delta p=2 m V_x \frac{n}{2} V_x \Delta t S$ than ...
The $L\cdot\Omega$ term comes directly from the change of frame of reference, especially from the transformation from the static frame of reference to the rotating frame of reference. Let $\mathcal{R}\equiv(x,y,z)$ the initial static frame, and $\widetilde{\mathcal{R}}\equiv(x',y',z')$ the rotating frame at a constant velocity ...