# Q. 14.20 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 [see Fig. 14.33].

Let the initial volume and pressure of the chamber be V and P.

Let the ball be pressed by a distance x.

This will change the volume by an amount  ax.

Let the change in pressure be $\Delta P$

Let the Bulk's modulus of air be K.

$\\K=\frac{\Delta P}{\Delta V/V}\\ \Delta P=\frac{Kax}{V}$

This pressure variation would try to restore the position of the ball.

Since force is restoring in nature displacement and acceleration due to the force would be in different directions.

$\\F=a\Delta P\\ -m\frac{\mathrm{d^{2}}x }{\mathrm{d}t^{2}}=a\Delta p\\ \frac{\mathrm{d^{2}}x }{\mathrm{d}t^{2}}=-\frac{ka^{2}}{mV}x$

The above is the equation of a body executing S.H.M.

The time period of the oscillation would be

$T=\frac{2\pi }{a}\sqrt{\frac{mV}{k}}$

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