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.

Explanation

Consider an air chamber of volume V with a long nech of uniform area of cross-section A, and a frictioless ball of mass m fitted smoothly in the nech at position , C Fig. the pressure of air below the bal inside the chamber is equal to the atmospheric pressure. Increase the pressure on the ball by a little amount p, so that the ball is depressed to position D, where CD=y
There will be decreases in volume and hence increase in pressure of air inside the chamber . The decrease in volume of the air inside the chamber , `DeltaV=Ay`
Volumetric strain =Charge in volume/original volume
`=(DeltaV)/V=(Ay)/V`
`:.` Bulk modulus of electricity E, will be
E=stress (or increase in pressure ) /Volumetric strain
`=(-p)/(Ay//V)=(-pV)/(Ay)`
Here, negative sign shows that the increase in pressure with decrease the volume of air in the chamber.
Now, `p=(-EAy)/V`
Due to thisi excess pressure, the restoring force acting on the ball is
`F=pxxA =(-EAy)/V. A =(-EA^(2))/V y....(i)`
Since `F propy` and negative sign show that the force is directed towards equilibrium position. if the applied increased pressure is removed from the ball , the ball will start executing linear SHM in the nech of chamber with C as mean position .
In S.H.M , the restoring force,
F=-ky
comparing (i) and (ii),. we have
spring factor, `k=EA^(2)//V`
Here, inertia factor =mass of ball =m
period , `T=2pisqrt(("inertia factor")/("spring factor"))`
`=2pisqrt(m/(EA^(2)//V))=(2pi)/A, sqrt((mV)/E)`
`:.` Frequency , v=`1/T =A/(2pi)sqrt(E/(mV))`
Note: if the ball osillates in the nech of chamber under isothermal conditions , thru, E=P=picture of air inside the chamber , when ball is at equilibrium position . if the ball oscillate in the neck of chamber under adiabatic conditions, then E=gP. where `g=C_(P)//C_(v)`