Ncert Exercises & Solutions

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.14.27). 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. 14.27].

Fig. 14.27

The volume of the air chamber = V
Area of the 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.
The decrease in the volume of the air chamber, ΔV = ax
Volumetric strain = $\frac{Change in volume}{Original volume}$

$\Rightarrow \frac{∆ V}{V} = \frac{ax}{V}$
$Bulk Modulus of the air, B = \frac{Stress}{Strain} = \frac{- ∆ P}{\frac{ax}{V}}$
$In this case, stress is equal to the increase in the pressure .$
$The negative sign indicates that pressure increases with a decrease in volume .$
$∆ P = \frac{- Bax}{V}$
$The restoring force acting on the ball,$
$F = ∆ P \times a$
$= \frac{- Bax}{V} . a$
$= \frac{- {Ba}^{2} x}{V}$
$Comparing it with the standard equation of SHM :$
$F = - kx, where k is the spring constant .$
$k = \frac{{Ba}^{2}}{V}$
$Time period,$
$T = 2 π \sqrt{\frac{m}{k}}$
$= 2 π \sqrt{\frac{Vm}{{Ba}^{2}}}$

NEET Information

courses

Company

Botany Questions

Chemistry Questions

Physics Questions

Zoology Questions