An air champer 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. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillation assuming pressure volume variations of air to to be isothermal.

Explanation

Let E be the bulk modulus of the material of the ball. Then,

`E=(-PV)/(DeltaV)`
`"or "P=(-E DeltaV)/(V)" but "F=PA`
`"thereby, "F=(-E DeltaV)/(V)A.` Let .y. be the displacement, Then `DeltaV=Ay.`
`"Hence, "F=-E(A^(2)y)/(V).` However using the Newton.s II Law of motion,
`F=ma =m(d^(2)y)/(dt^(2))`
Hence `m(d^(2)y)/(dt^(2))=-(EA^(2)y)/(V)`
`"or "(d^(2)y)/(dt^(2))+((EA^(2))/(mV))y=0`
Comparing this with `(d^(2)y)/(dt^(2))+omega^(2)y=0`, we get,
`omega^(2)=(EA^(2))/(mV)" so that " (4pi^(2))/(T^(2))=(EA^(2))/(mV)`
`or T=((2pi)/(A))sqrt((mV)/(E))`.