Question

One metre long wire is fixed between two rigid supports. The tension is the wire is $200 \,N$ and mass per unit length of the wire is $\mu=2x$, where $x$ is the distance from one end of the wire. Find the time (in second) the pulse takes to reach the other end of the wire

Answer 1

The pulse velocity is given by,
$v=\sqrt{\frac{F}{\mu}}=\sqrt{\frac{200}{2x}}=\frac{10}{\sqrt{x}} $
or $\frac{dx}{dt} = \frac{10}{\sqrt{x}}$
or $\int\limits_{0}^{1} \sqrt{x}dx=10\int\limits_{0}^{t}dt$
or $ \int\limits_{0}^{1} x^{1 2}dx = 10\int\limits_{0}^{t} dt $
or $\left|\frac{x^{3 /2}}{3/ 2}\right|_{0}^{1} =10t $
or $t=\frac{2}{30} s =0.067\,s$