Question

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 ** of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.33].

Answer 1

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 = Change in volume/ Original volume

⇒ ΔV/V = ax/V

Bulk Modulus of air B = Stress/Strain = -p/ax/V

In this case, stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume.