First slide

Stationary waves

Question

A stone is hung in air from a wire which is stretched over a sonometer. The bridges of the sonometer are L cm apart when the wire is in unison with a tuning fork of frequency N. When the stone is completely immersed in water, the length between the bridges is l cm for re-establishing unison, the specific gravity of the material of the stone is

Moderate

A

$\frac{L^{2}}{L^{2} + l^{2}}$
B

$\frac{L^{2} - l^{2}}{L^{2}}$
C

$\frac{L^{2}}{L^{2} - l^{2}}$
D

$\frac{L^{2} + l^{2}}{L^{2}}$

Solution

Frequency of vib. is stretched string $n = \frac{1}{2 (Length)} \sqrt{\frac{T}{m}}$
When the stone is completely immersed in water, length changes but frequency doesn’t ( $∵$ unison reestablished)
Hence length $\propto \sqrt{T} \Rightarrow \frac{L}{l} = \sqrt{\frac{T_{air}}{T_{water}}} = \sqrt{\frac{Vρg}{V (ρ - 1) g}}$
(Density of stone = $ρ$ and density of water =1)
$\Rightarrow \frac{L}{l} = \sqrt{\frac{ρ}{ρ - 1}} \Rightarrow ρ = \frac{L^{2}}{L^{2} - l^{2}}$

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