Question

A rope of length $L$ and mass $M$ hangs freely from the ceiling. If the time taken by a transverse wave to travel from the bottom to the top of the rope is $T$, then time to cover first half length is

• A. $T$
• B. $T\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)$
• C. $\frac{T}{\sqrt{2}}$
• D. $\frac{T}{2}$

Answer 1

Answer: (C)

$v=\sqrt{\frac{N}{\mu }}$
The tension $N$ in the string varies as:
$N=\pi \frac{Mg}{L}\times x$ where $x$ is length from the ground.
$dt=\frac{dx}{v_x}$ and $v_x=\sqrt{\frac{Mgx}{L\times M/L}}=\sqrt{gx}$
$\underset{0}{\overset{\top }{\int }}dt=\underset{0}{\overset{L}{\int }}\frac{dx}{\sqrt{gx}}$
$T=\underset{0}{\overset{L}{\int }}2\sqrt{x}dx$
$T=\underset{0}{\overset{L}{\int }}2\sqrt{L_g}$...(i)
If time to cover half length is $T_2$.
$T_2=\sqrt{2Lg}$ [By putting limits 0 to $L/2$ in equation (i)]
$\frac{T}{\sqrt{2}}=T_2$