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
- $\frac{1}{2 \pi} \frac{A_{\gamma}p_0}{V_0M}$
- $\frac{2}{2\pi}\frac{V_0MP_0}{A^2 \gamma}$
- $\frac{2}{2\pi}\sqrt{\frac{A^2\gamma P_0}{MV_0}}$
- $\frac{1}{2 \pi} \sqrt{\frac{MV_0}{A_{\gamma}P_0}}$
The Correct Option is C
Solution and Explanation
$ \, \, \, \, \, \, \, \, \, p_0A =Mg \, \, \, \, \, \, \, \, \, \, \, \, $ ...(i)
when slightly displaced downwards.
$dp=-\gamma\bigg(\frac{p_0}{V_0}\bigg)dV\bigg( As \, in \, adiabatic \, process , \frac{dp}{dV}=-\gamma \frac{p}{V}\bigg)$
$\therefore \, \, $ Restoring force,
$ \, \, \, \, \, \, \, \, \, \, \, \, F=(dp)A=-\bigg(\frac{\gamma p_0}{v_0}\bigg)(A)(Ax)$
$ \, \, \, \, \, \, \, \, \, \, F ? -x$
Therefore, motion is simple harmonic comparing with
F = - kx we have
$ \, \, \, \, \, \, \, \, \, k=\frac{\gamma p_0A^2}{V_0}$
$\therefore \, \, \, \, \, \, f=\frac{1}{2}\pi \sqrt{\frac{k}{m}}=\frac{1}{2 \pi} \sqrt{\frac{\gamma p_0A^2}{MV_0}}$
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Concepts Used:
Oscillations
Oscillation is a process of repeating variations of any quantity or measure from its equilibrium value in time . Another definition of oscillation is a periodic variation of a matter between two values or about its central value.
The term vibration is used to describe the mechanical oscillations of an object. However, oscillations also occur in dynamic systems or more accurately in every field of science. Even our heartbeats also creates oscillations. Meanwhile, objects that move to and fro from its equilibrium position are known as oscillators.
Read More: Simple Harmonic Motion
Oscillation- Examples
The tides in the sea and the movement of a simple pendulum of the clock are some of the most common examples of oscillations. Some of examples of oscillations are vibrations caused by the guitar strings or the other instruments having strings are also and etc. The movements caused by oscillations are known as oscillating movements. For example, oscillating movements in a sine wave or a spring when it moves up and down.
The maximum distance covered while taking oscillations is known as the amplitude. The time taken to complete one cycle is known as the time period of the oscillation. The number of oscillating cycles completed in one second is referred to as the frequency which is the reciprocal of the time period.
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