Question

A uniform rope of mass $0.1kg$ and length $2.45m$ hangs from a ceiling. The time taken by a transverse wave to travel the full length of the rope is $\left(g=9.8m/s^2\right)$

• A. 1 s
• B. 2 s
• C. 3 s
• D. 4 s

Answer 1

Answer: (A)

As the rope has mass and it is suspended vertically, tension in it will be different at different points. For a point at a distance $x$ from the free end, tension will be due to the weight of the rope below it. If $M$ is the mass of rope of length $L$, the mass of length $x$ of the rope will be $(M/L)x$.
$\therefore T=\left[\frac{M}{L}x\right]g$
and $v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{Mgx}{L(M/L)}}=\sqrt{gx}...(i)$
The tension and the velocity of the wave is different at different points. If at point $x$ the wave travels a distance $dx$ in time $dt$,
$v=\frac{dx}{dt}\text{ or }\sqrt{gx}=\frac{dx}{dt}$ (Using(i))
or $\int dt=\int \frac{dx}{\sqrt{gx}}$ or $t=\frac{1}{\sqrt{g}}\int {}_0^L{x}^{-1/2}dx$
or $t=2\sqrt{\left(\frac{L}{g}\right)}$
As $L=2.45m$
$\therefore t=2\sqrt{\frac{2.45}{9.8}}=1s$