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

Answer 1

Answer: (A)

For isothermal process

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