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Sonometer

Question

A uniform rope of length L and mass $m_{1}$ hangs vertically from a rigid support. A block of mass $m_{2}$ is attached to the free end of the rope. A transverse pulse of wavelength $λ_{1}$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is $λ_{2}$ . The ratio $\frac{λ_{2}}{λ_{1}}$ is

Moderate

A

$\sqrt{\frac{m_{1}}{m_{2}}}$
B

$\sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$
C

$\sqrt{\frac{m_{2}}{m_{1}}}$
D

$\sqrt{\frac{m_{1} + m_{2}}{m_{1}}}$

Solution

$T_{1} = m_{2} g$
$T_{2} = (m_{1} + m_{2}) g$
Velocity $\propto \sqrt{T}$
$λ \propto \sqrt{T}$
$\frac{λ_{1}}{λ_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}}$
$\Rightarrow \frac{λ_{2}}{λ_{1}} = \sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$

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