Question

A nonuniform string of mass $45kg$ and length $1.5m$ 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 $15N$ which is uniform. Find the time (in second) required for a pulse generated at one end of the string to travel to the other end.

Answer 1

Here, $\mu =\frac{dM}{dx}=kx$
${\int }_0^{M}dM=k{\int }_0^{l}xdx$
$M=k\frac{l^2}{2}\Rightarrow k=\frac{2M}{l^2}....\left(1\right)$
The velocity of wave in the string is $v=\sqrt{\frac{T}{\mu }}$
$\frac{dx}{dt}=\sqrt{\frac{T}{kx}}$
or ${\int }_0^l\sqrt{x}dx=\sqrt{\frac{T}{k}}{\int }_0^{t}dt$
or $t=\sqrt{\frac{k}{T}}\frac{l^{3/2}}{3/2}=\frac{2}{3}\times \sqrt{\frac{2Ml}{T}}$ using (1)
or $t=\frac{2}{3}\times \sqrt{\frac{2\times 45\times 1.5}{15}}=\frac{2}{3}\times 3=2s$