Chapter 18 - Exercises and Problems - Page 332: 70

$\omega = \frac{AP_0}{\sqrt{MnRT_0}}$

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: $\omega = \frac{AP_0}{\sqrt{MnRT_0}}$

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