Question

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.14.33). 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. 14.33].

Answer 1

Let the ball is depressed by x units . As a result the volume will decrease and this would increase the pressure inside .
decrease the volume of air ( ∆V) = cross section area of neck( a) × depression length = ax
∆V = ax

Volume strain = ∆V/V = ax/V
We know,
Bulk modules = -P/(∆V/V)
[ becoz volume stress = bulk stress = P]
B = - P/ax/V
P = -Bax/V
Also we know, F/a = P [ pressure = force/area ]
F/a = - Bax/V
F = -Ba²x/V
Compare with F = -mw²x
w = √(Ba²/mV)
T = 2π√(mV/Ba²)

Answer 2

hope you understood! have a good day !