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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]. |
Answer» Solution :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))` |
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