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 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 $2\pi m^x{V}^y{B}^z{a}^q$. The value of $\frac{xy}{q}$ is_______.

Answer 1

Answer: -0.25

Change in volume, $\Delta V=ax$
excess pressure, $\Delta P=-B\frac{\Delta V}{V}$
Restoring force $=$ excess pressure
$\times$ cross-sectional area
$\Rightarrow F=-Ba\frac{\Delta V}{V}$
$=\frac{-B{a}^2}{V}x[\because \Delta V=ax]$
In SHM, F = $-kx$
$\therefore k=\frac{B{a}^2}{V}$
$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{mV}{B{a}^2}}$
$\Rightarrow x=\frac{1}{2},y=\frac{1}{2},z=\frac{-1}{2},q=-1$
$\Rightarrow \frac{xy}{q}=\frac{\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)}{(-1)}=-0.25$