First slide

Applications of SHM

Question

An air chamber of volume V, has a long of cross-sectional area A. A ball of mass m is fitted smoothly in the neck. The bulk modulus of air is B. If the ball is pressed down slightly and released, the time period of its oscillation is

Difficult

A

$2 π \sqrt{\frac{m V}{2 B A^{2}}}$
B

$π \sqrt{\frac{2 m V}{B A^{2}}}$
C

$2 π \sqrt{\frac{m V}{B A^{2}}}$
D

$\frac{π}{2} \sqrt{\frac{m}{B A^{2}}}$

Solution

Let the ball is pressed down by a small amount y, then the value of air decreases by Ay.
Then, excess pressure is $d p = - B (\frac{d V}{V}) = - B (\frac{y A}{V})$
A restoring force F(Adp) acts in the upward direction.
$\begin{array}{l} ∴ F = A [- B (\frac{y A}{V})] = - \frac{B A^{2}}{V} \cdot y \\ ∴ a = - \frac{B A^{2}}{m V} y comparing this with a = - ω^{2} y \\ \Rightarrow ω = \sqrt{\frac{B A^{2}}{m V}} \\ \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m V}{B A^{2}}} \end{array}$

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