Question

An air chamber of volume V has a neck of cross-sectional area a into which a light ball of mass m just fits and can move up and down without friction. The diameter of the ball is equal a to that of the neck of the chamber. The ball is pressed down a little and released. If the bulk modulus of air is B, the time period of the oscillation of the ball is

• A. $\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { Ba } ^ { 2 } } { \mathrm { mV } } }$
• B. $\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { BV } } { \mathrm { ma } ^ { 2 } } }$
• C. $\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { mB } } { \mathrm { Va } ^ { 2 } } }$
• D. $\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { mV } } { \mathrm { Ba } ^ { 2 } } }$

Answer 1

Answer: (D)

$\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { mV } } { \mathrm { Ba } ^ { 2 } } }$

Answer 2

The situation is as shown in the figure. Let P be pressure of air in the chamber. When the ball is presses down a distance x, the volume of air decrease from V to say V $\Delta$P the change in volume is

The excess pressure $\Delta \mathrm { P }$ is related to the bulk modulus B as

$\Delta \mathrm { P } = - \mathrm { B } \frac { \Delta \mathrm { V } } { \mathrm { V } }$

Restoring force on ball = excess pressure × cross sectional area

Or

Or $\mathrm { F } = - \frac { \mathrm { Ba } ^ { 2 } } { \mathrm {~V} } \mathrm { x } \quad ( \because \Delta \mathrm { V } = \mathrm { ax } )$

Or $\mathrm { F } = - \mathrm { kx }$

Where $\mathrm { k } = \frac { \mathrm { Ba } ^ { 2 } } { \mathrm {~V} }$

i.e. $F \propto - x$

hence the motion of the ball is simple harmonic if m is the ball the time period of the SHM is

$\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm {~m} } { \mathrm { k } } }$ or $\mathrm { T } = 2 \pi \sqrt { \frac { \mathrm { mV } } { \mathrm { Ba } ^ { 2 } } }$