Answer
$\omega = \frac{AP_0}{\sqrt{MnRT_0}}$
Work Step by Step
We know the following equations:
$\omega = \frac{\Delta d}{T_p}=\frac{2\Delta x}{T_p}$
Where $T_p$ is the time it takes to complete a period of motion.
We also know that the ideal gas law can be simplified to:
$V = \frac{nRT}{P}$
In addition, we know that the change in volume can be given by:
$\Delta V = A \Delta x $
As stated in example 18.3, this process of the pistons moving is adiabatic, so it follows:
$PV^r = P_0V_0^r$
In addition, we know from Newton's second law that $F=ma$, and we know that $a=\frac{d\Delta x^2}{d^2t}$
Combining these equations, we obtain:
$\omega = \frac{AP_0}{\sqrt{MnRT_0}}$
