Question

A non-uniform wire of length L and mass M has a variable linear mass density given by $\mu =kx,$ where x is distance form one end of wire and k is a constant. Find the time taken by a pulse starting at one end to reach the other end when the tension in wire is T.

• A. $\frac{5}{3}\sqrt{\frac{ML}{T}}$
• B. $\frac{2}{3}\sqrt{\frac{2ML}{T}}$
• C. $\frac{8}{3}\sqrt{\frac{ML}{T}}$
• D. $\frac{3}{2}\sqrt{\frac{2ML}{T}}$

Answer 1

The correct option is B $\frac{2}{3}\sqrt{\frac{2ML}{T}}$

Total mass of the wire=$M={\int }_0^L\mu dx={\int }_0^{L}kxdx=k\frac{L^2}{2}$

$⟹k=\frac{2M}{L^2}$

Speed of pulse in the wire=$\sqrt{\frac{T}{\mu }}=\sqrt{\frac{T}{kx}}$

Thus time taken to reach from one end to other=${\int }_0^L\frac{dx}{\sqrt{\frac{T}{kx}}}$

since $k=\frac{2M}{L^2}$,

time taken=$\frac{2}{3}\sqrt{\frac{2ML}{T}}$