Question

An air chamber having a volume V and a cross-sectional area of the neck is $a$ into which a ball of mass $m$ can move up and down without friction. 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.

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, $△V=ax$
Volumetric strain $=$ Change in volume/original volume

$\Rightarrow △V/V=ax/V$
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 decrease in volume.

$p=-Bax/V$
The restoring force acting on the ball, $F=p\times a$

$=-Bax/V\times a$

$=-B{a}^{2}x/V$ ...(i)
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:

$k=B{a}^2/V$

Time Period,

$\therefore T=2\pi \sqrt{\frac{m}{k}}$

$=2\pi \sqrt{\frac{Vm}{B{a}^2}}$