Question

A flexible steel cable of total length L and mass per unit length $\mu$ hangs vertically from a support at one end. How long will it take for a wave to travel down the cable ?

• A. $t=4\sqrt{\frac{L}{g}}$
• B. $t=4\sqrt{\frac{L}{g-a}}$
• C. $t=2\sqrt{\frac{L}{g}}$
• D. $t=9\sqrt{\frac{L}{g+a}}$

Answer 1

The correct option is C $t=2\sqrt{\frac{L}{g}}$

Tension at a point at a distance $x$ from the support is the due to the weight of the cable below it.

Let $m$ be the mass of the cable.

Thus $T=m\frac{(L-x)}{L}g$

Mass per unit length, $\mu =\frac{m}{L}$

Thus speed of the wave at a given point=$\sqrt{\frac{T}{\mu }}=\sqrt{\frac{m\frac{(L-x)}{L}g}{m/L}}=\sqrt{g(L-x)}$

Velocity of wave at a distance $x$ from support=$\sqrt{g(L-x)}$

Time taken to transverse a distance element of $dx,dt=\frac{dx}{\sqrt{g(L-x)}}$

Thus total time taken=${\int }_0^L\frac{dx}{\sqrt{g(L-x)}}$

$=2\sqrt{\frac{L}{g}}$