Question

A rectangular pipe having cross-sectional area $A$ . It is closed at one end and at its other end, a block having the same cross-section and mass $m$ is placed such that the system is airtight. In the equilibrium position of the block, the pressure and volume of air enclosed in the pipe are $P$ and $V$ respectively. If the block is displaced by a small distance $x$ inward and released then find the time period of S.H.M. [Assume the walls are frictionless and compression of air is isothermal].

• A. $T=2\pi {\left(\frac{mV}{P{A}^2}\right)}^{1/2}$
• B. $T=\frac{\pi }{2}{\left(\frac{mV}{P{A}^2}\right)}^{1/2}$
• C. $T=\pi {\left(\frac{P{A}^2}{mV}\right)}^{1/2}$
• D. $T=2\pi {\left(\frac{2mv}{P{A}^2}\right)}^{1/2}$

Answer 1

Answer: (A)

For isothermal process

$PV=C$
$PdV+Vdp=0$
$dp=\frac{-P}{V}\cdot dV$
$dp=\frac{-p}{V}\left(Ax\right)$
$\Rightarrow F_{net}=(dP)A=-\left(\frac{P{A}^2}{V}\right)x$
$\Rightarrow F=-kx$