JEE Mains · Physics · STD 11 - 13. oscillations
An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass \(M\). The piston and the cylinder have equal cross sectional area \(A\). When the piston is in equilibrium, the volume of the gas is \(V_0\) and its pressure is \(P_ 0\). The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
- A \(\frac{1}{{2\pi }}\sqrt {\frac{{M{V_0}}}{{A\gamma {P_o}}}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\)
- B \(\;\frac{1}{{2\pi }}\sqrt {\frac{{A\gamma {P_o}}}{{{V_0}M}}} \)
- C \(\;\frac{1}{{2\pi }}\sqrt {\frac{{{A^2}\gamma {P_o}}}{{M{V_0}}}} \)
- D \(\;\frac{1}{{2\pi }}\frac{{{V_o}M{P_o}}}{{{A^2}\gamma }}\)
Answer & Solution
Correct Answer
(C) \(\;\frac{1}{{2\pi }}\sqrt {\frac{{{A^2}\gamma {P_o}}}{{M{V_0}}}} \)
Step-by-step Solution
Detailed explanation
\({\frac{M g}{A}=P_{0}}\) \( {P_{0} V_{0}^{\gamma}=P V^{\gamma}}\) \({\mathrm{Mg}=\mathrm{P}_{0} \mathrm{A}}{\ldots(1)}\) \( {P_{0} A x_{0}^{\gamma}=P A\left(x_{0}-x\right)^{\gamma}}\) \(P=\frac{P_{0} x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\) Let piston is displaced by…
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