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:
$N=\pi \frac{M g}{L} \times x$ where $x$ is length from the ground.
$d t=\frac{d x}{v_{x}}$ and $v_{x}=\sqrt{\frac{M g x}{L \times M / L}}=\sqrt{g x}$
$\int\limits_{0}^{\top} d t=\int\limits_{0}^{L} \frac{d x}{\sqrt{g x}}$
$T=\int\limits_{0}^{L} 2 \sqrt{x} d x$
$T=\int\limits_{0}^{L} 2 \sqrt{L_{g}}$...(i)
If time to cover half length is $T_{2}$.
$T_{2}=\sqrt{2 L g}$ [By putting limits 0 to $L / 2$ in equation (i)]
$\frac{T}{\sqrt{2}}=T_{2}$