Question

A uniform rope of mass $0.1\text{kg}$ and length $2.45\text{m}$ hangs from a ceiling. The time taken by a transverse wave to travel through the full length of the rope is
[Take $g=9.8{\text{m/s}}^2$]

• A. $2.0\text{s}$
• B. $1.2\text{s}$
• C. $1.0\text{s}$
• D. $2.2\text{s}$

Answer 1

The correct option is C $1.0\text{s}$

Let $M$ be the total mass of the rope and $L$ be its length.


Tension at distance $x$ from the bottom is given by
$T=\left(\frac{M}{L}x\right)g$

Velocity of wave at distance $x$, $v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{\left(\frac{Mx}{L}\right)g}{\frac{M}{L}}}=\sqrt{gx}$

$\Rightarrow v=\frac{dx}{dt}=\sqrt{gx}$ or $dt=\frac{dx}{\sqrt{gx}}$
Integrating from $x=0$ to $L$,
$t=\frac{1}{\sqrt{g}}\underset{0}{\overset{L}{\int }}{x}^{-\frac{1}{2}}dx$
or $t=2\sqrt{\frac{L}{g}}=2\sqrt{\frac{2.45}{9.8}}=1\text{s}$