Question

A uniform rope of mass $0.1kg$ and length $2.45m$ hangs from a rigid support. the time taken by the transverse wave formed in the rope to travel through the full length of the rope is (Assume $g=9.8m/s^2)$:

• A. $0.5s$
• B. $1.6s$
• C. $1.2s$
• D. $1.0s$

Answer 1

The correct option is D $1.0s$

Tension in the string at a distance $x$ from the bottom is given by: $T_x=\frac{Mx}{L}g$

Speed of wave is given by, $v=\sqrt{\frac{T}{\mu }}=\sqrt{xg}$, where $\mu =\frac{M}{L}=$ mass per unit length

Now, $v=\frac{dx}{dt}=\sqrt{xg}$

$dt=\frac{1}{\sqrt{g}}{\int }_0^L{x}^{-1/2}dx$

$⟹t=2\sqrt{\frac{L}{g}}=1s$