Question

A non-uniform rope of mass $M$ and length $L$ has a variable linear mass density given by $\mu =kx$, where $x$ is the distance from one end of the wire and $k$ is a constant. The time required for a pulse generated at one end of the wire to travel to the other end is given by $t=\sqrt{\left(pML/9F\right)}$ where $F$ (constant) is the tension in the wire then find $p$.

Answer 1

Linear mass density

$\mu =kx$

Now, velocity $v=\sqrt{\frac{T}{\mu }}$

$\Rightarrow v=\sqrt{\frac{T}{kx}}$

Time taken in travelling the $dx$ distance $x$

$dt=\frac{dx}{v}$

$\Rightarrow dt=\sqrt{\frac{kx}{T}}dx$

Integrating on both side,

$t={\int }_0^l\sqrt{\frac{kx}{T}}dx$

$\Rightarrow t=\sqrt{\frac{K}{T}}{\int }_0^l\sqrt{x}dx$

$\Rightarrow t=\sqrt{\frac{K}{T}}{\left[\frac{2x^{3/2}}{3}\right]}_0^l$

$\Rightarrow t=\frac{2}{3}\sqrt{\frac{k}{T}}{l}^{3/2}$

$\Rightarrow t=\sqrt{\frac{4k{l}^3}{9T}}$

$\Rightarrow t=\sqrt{\frac{4k{l}^3}{9F/A}}$

$\Rightarrow t=\sqrt{\frac{4Ml}{9F}}$

$\therefore p=4$