Question

A non uniform string of mass $45\text{kg}$ and length $1.5\text{m}$ has a variable linear mass density given by $\mu =Kx$, where $x$ is the distance from one end of the string and $K$ is a constant. Tension in the string is $15\text{N}$ (uniform). Find the time (in seconds) required for a pulse generated at one end of the string to travel to the other end.

Answer 1

Given that
Linear mass density $\left(\mu \right)=Kx=\frac{dM}{dx}$
where, $M=$ Total mass of string
$\Rightarrow {\int }_0^{M}dM={\int }_0^{l}Kxdx$
$\Rightarrow M=\frac{K{l}^2}{2}$ [$l=$ length of string]
$\Rightarrow K=\frac{2M}{l^2}$

Wave velocity, $V=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{T}{Kx}}=\sqrt{\frac{T}{K}}\frac{1}{\sqrt{x}}$
$\Rightarrow \frac{dx}{dt}=\sqrt{\frac{T}{K}}\frac{1}{\sqrt{x}}[\because V=\frac{dx}{dt}]$

To find time taken for the pulse to travel from end to end, we have to integrate the equation above with limits of $x$ from $0$ to $L$.
$\Rightarrow {\int }_0^l\sqrt{x}dx={\int }_0^t\sqrt{\frac{T}{K}}dt$
$\Rightarrow {\left[\frac{2}{3}{x}^{\frac{3}{2}}\right]}_0^l=\sqrt{\frac{T}{K}}[t{]}_0^t$
$\Rightarrow t=\sqrt{\frac{K}{T}}\times \frac{2}{3}{l}^{\frac{3}{2}}$
$=\frac{2}{3}\sqrt{\frac{K{l}^3}{T}}=\frac{2}{3}\sqrt{\frac{2Ml}{T}}[\because k{l}^2=2M]$
$\therefore t=\frac{2}{3}\sqrt{\frac{2\times 45\times 1.5}{15}}=2\text{sec}$