Question
An ideal gas is enclosed in a horizontal cylinder of area of cross-section A fitted with an airtight and frictionless piston of mass M. The piston is in equilibrium with atmospheric pressure P. The volume of gas enclosed in the cylinder is V. The cylinder is slightly displaced from equilibrium position and released. Show that the motion of piston is simple harmonic motion and find the time period of oscillation. Assume that the system is completely isolated from surrounding.
Solution
The system is as shown in the figure below.
Let the piston be displaced by y towards left.
Let dV be the decrease in volume and dP be the increase of pressure.
The system is isolated from surrounding. Therefore, the change in pressure and volume is according to adiabatic conditions.
Thus the equation of state can be written as,
The equation (1) gives the change in pressure of gas when the volume of gas is changed by dV.
Here, change in the volume of gas when the piston is displaced by y is Ay.
Thus,
Force on piston is,
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The acceleration of piston is,
So, it is clear from the above equation that the acceleration is proportional to displacement and directed towards the mean position.
Therefore the motion of piston is simple harmonic motion.
The time period of oscillation is,
Let the piston be displaced by y towards left.
Let dV be the decrease in volume and dP be the increase of pressure.
The system is isolated from surrounding. Therefore, the change in pressure and volume is according to adiabatic conditions.
Thus the equation of state can be written as,
The equation (1) gives the change in pressure of gas when the volume of gas is changed by dV.
Here, change in the volume of gas when the piston is displaced by y is Ay.
Thus,
Force on piston is,
The acceleration of piston is,
So, it is clear from the above equation that the acceleration is proportional to displacement and directed towards the mean position.
Therefore the motion of piston is simple harmonic motion.
The time period of oscillation is,
