Question

An air chamber of volume V has a neck area of cross-section into which a ball of mass m just fits and can move up and down without any 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 = V 

mass of ball = m 

cross-sectional Area = A 

initial pressure on either side of the ball = atmospheric pressure = P 

Let the charge in volume of air when ball is pressed be ∆V 

∆V = Ay (y = displacement) 

Now, Bulk modulus of elasticity

The above equation is of the form,

Hence it is in SHM.