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An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction (Fig.14.33). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal .
Asked by Praseem Last Modified
Answer Volume of the air chamber = V
Area of cross-section of the neck = a
Mass of the ball = m
The pressure inside the chamber is equal to the atmospheric pressure.
Let the ball be depressed by x units. As a result of this depression, there would be a decrease in the volume and an increase in the pressure inside the chamber.
Decrease in the volume of the air chamber, ΔV = ax
Volumetric strain 
Bulk Modulus of air,
In this case, stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume.
The restoring force acting on the ball,
F = p × a
In simple harmonic motion, the equation for restoring force is:
F = –kx … (ii)
Where, k is the spring constant
Comparing equations (i) and (ii), we get:
Time period,
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