Question

A string of length L and mass M hangs freely from a fixed point. Calculate
(a) The velocity of the transverse wave long the string at any position.
(b) Time taken by a transverse pulse to traverse the string.

Answer 1

Mass per unit length of the string $=\frac{M}{L}$ ..(i)
Let there be a point on the string at a distance x from the free end.
Tension at the point
T$=$(weight of the string per unit length $\times$ part of the spring)
$T=\frac{M}{L}x.g$
Hence, velocity of transverse wave along the string,
$v=\sqrt{\frac{T}{m}}=\sqrt{\frac{\left(Mgx\right)/L}{M/L}}=\sqrt{gx}$ (ii)
(b) From equation (ii) we have
$v=\sqrt{gx}$
$\Rightarrow \left(\frac{dx}{dt}\right)=\sqrt{gx}$ or $dt=\frac{dx}{\sqrt{gx}}$
$\therefore$ Required time $t={\int }_0^L\frac{dx}{\sqrt{gx}}=\frac{1}{\sqrt{g}}{\int }_0^L{x}^{-1/2}dx$
$t=2\sqrt{\frac{L}{g}}$.