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

Hint: In the above given diagram it is given that the system is Isothermal that means the temperature in the whole system is assumed to be constant. Obtain the formula for the restoring force on the ball and the restoring force in simple harmonic motion. Compare both forces to obtain a time period of oscillations.

Complete step by step answer:
Let us consider the volume of the air chamber is $V$
Assume the area of cross-section of the neck is $a$
Also let the value of the Mass of the ball is $m$
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, $\Delta V = ax$
Write the expression for the Bulk modulus of air.
$B = \dfrac{-p}{\dfrac{ax}{V}}$
In this case Stress is the increase in the pressure.
The negative sign indicates that it increases with decrease in Volume.
Now, write the expression for the pressure.
$P = \dfrac{-Bax}{V}$
The restoring force acting on the ball is calculated as,
$F = P \times a = \left( { - Bax/V} \right)\left( a \right) = - B{a^2}x/V$.........…. (i)
We know that in Simple Harmonic Motion, the equation for restoring force is written as
$F = - kx$.........…. (ii)
 Where, $k$ is the spring constant.
 Comparing equation (i) and (ii) we get,
$ - kx = - \dfrac{{B{a^2}x}}{V}$
$\Rightarrow k = \dfrac{B{a^2}}{V} $
Time period for the equation is,
$T = 2\pi \sqrt {\dfrac{m}{k}} $
Substitute the values of $k$ in the above equation to obtain,
$T = 2\pi \sqrt{\dfrac{mV}{B{a^2}}}$

Note: Do not confuse the negative sign mentioned in the expression. The negative sign indicates decrease in the volume and an increase in the pressure inside the chamber.