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].

Explanation

Consider an air chamber of volume V with a long neck of uniform area of cross-section a, and a frictionless ball of mass m fitted smoothly in the neck at position C shown in figure. The pressure of air below the ball inside the chamber is equal to the atmosphere pressure.

Increase the pressure on the ball by a little amoung P, so that ball is depressed to position D, where CD = y
There will be decrease in volume `triangleV = Ay`
Volume strain `=(" Change in volume ")/(" Original volume ")`
`= (triangle V)/(V) = (Ay)/(V)`
Bulk Modulus `B= (" Stress ")/(" Volumetric strain ")`
`therefore B = (-P)/(Ay"/"V) = -(PV)/(Ay)`
Here, negative sign shows that the increase in pressure will decrease the volume
`therefore P= (-BAy)/(V)`
Due to this excess pressure the restoring force acting on the ball is,
`F= PA = (-BAy)/(V)xx A`
`therefore F= -(BA^(2)y)/(V)`
`= -ky`
`therefore F propto -y`, where force constant `k= (BA^(2))/(V)`
The motion of a ball around C is SHM and from restoring force `F= -ky`,
`k= (BA^(2))/(V)`
Period `T= 2pi sqrt((m)/(k))`
`= 2pi sqrt((m)/((BA^2)/(V)))`
`= 2pi sqrt((mV)/(BA^2))`