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.

An air champer of volume 'V' has a neck area of cross - section A into

1.	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.
Answer» <P> Solution :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))`.