Question

A heavy but uniform rope of length $L$ is suspended from a ceiling. A particle is dropped from the ceiling. At the same instant, the bottom end is given a jerk. Find the velocity of wave pulse at an instant when particle and pulse are at same level.

• A. $\sqrt{\frac{gL}{2}}$
• B. $\sqrt{\frac{gL}{5}}$
• C. $\sqrt{\frac{gL}{3}}$
• D. $\sqrt{\frac{gL}{4}}$

Answer 1

The correct option is C $\sqrt{\frac{gL}{3}}$

Let after time $t$ from start, pulse meets the particle at distance $y$ from lower end. $t={\int }_0^y\frac{1}{\sqrt{gx}}dx=\sqrt{\frac{4y}{g}}$

Distance travelled by particle in this time is $(L-y)$

$u=0;a=g\therefore S=ut-\frac{1}{2}g{t}^2$

$L-y=ut-\frac{1}{2}g\times (\sqrt{\frac{4y}{g}}{)}^2$

$L-y-0-2y=0$

$L=3y\Rightarrow y=\frac{L}{3}$

$v_{pulse}=\sqrt{gy}$

$v_{pulse}=\sqrt{g\cdot \frac{L}{3}}$