InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
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» 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)` |
|