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, figure. 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

For isothermal process
`PV =` constant
`(dP)/(dA A) = -(P)/(AA)`
Bulk modulus `B = -(VdP)/(dAA)`
`rArr B = P` (isothermal bulk modulus)
Also `dP = - (Pdv)/(V)`
Restoring force `F = (dP)a = -(Pa)/(V)dV`
`dA A = a(dx)` where `dx` small vertical displacement of a ball
`F = -(Pa^(2))/(V)dx ....(1)`
`F = -kx .....(2)`
(force equation of `SHM`)
From equation `1` and `2`
`T = 2pisqrt((m)/(k))`
`T = 2pisqrt((Vm)/(Ba^(2)))`
`T = 2pisqrt((Vm)/(ba^(2)))` where `B` is the bulk modulus of air. for isothermal changes `B = P`.