Question

A piston of cross-section area $A$ is fitted in cylinder in which gas of volume $ V $ at pressure $ P $ is enclosed. Gas obeys Boyle's law, what is angular frequency it piston is displaced slightly?

• A. $ \sqrt{\frac{Ag}{V}} $
• B. $ 2\sqrt{\frac{Ag}{V}} $
• C. $ \sqrt{\frac{2Ag}{V}} $
• D. $ \frac{3Ag}{V} $

Answer 1

Answer: (A)

Since, gas obey’s Boyle’s law, so it is isothermal process.
The bulk modulus for gas is given by
$ B =- \frac {Δ p}{ΔV /V} $
But as gas obeys Boyle’s law,so
$pV =$ constant (isothermal process)
In isothermal process, isothermal bulk modulus of gas is equal to the pressure of the gas at that instant of time or
$B = p $
$∴ p = \frac{Δ p}{ΔV/ V} $
$⇒ Δ p = - \frac{p}{V} ΔV$
$⇒ \frac{ F}{A} = -\frac{p}{V} Ax$
$⇒ F = -\frac{pA^2}{V} x$
This equation is similar to
$F = - kx $
where $k = $ force constant of spring,
So, $k = \frac{ pA^2}{V} $
Hence, angular frequency
$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{pA^{2}}{mv}} $
$\sqrt{\frac{\left(mg\right)A}{mV}} \left(\because p = \frac{mg}{A}\right) $
$\omega = \sqrt{\frac{Ag}{V}}$