A uniform rope of length L and mass m1 hangs vertically from a rigid support. A block of mass m2is attached to the free end of the rope. A transverse pulse of wavelength λ1 is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ2. The ratio λ2λ1 is

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

A
$\sqrt{\frac{m_{2}}{m_{1}}}$
B
$\sqrt{\frac{m_{1} + m_{2}}{m_{1}}}$
C
$\sqrt{\frac{m_{1}}{m_{2}}}$
D
$\sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$

Solution:

$Wavelength of pulse at the lower$ $end (λ_{1}) \propto velocity (v_{1}) = \sqrt{\frac{T_{1}}{μ}}$

$Similarly, λ_{2} \propto v_{2} = \sqrt{\frac{T_{2}}{μ}}$

$∴ \frac{λ_{2}}{λ_{1}} = \sqrt{\frac{T_{2}}{T_{1}}} = \sqrt{\frac{(m_{1} + m_{2}) g}{m_{2} g}} = \sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$

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