Question

A uniform rope of length $l$ and mass $m$ hangs vertically from a rigid support. A block of mass $m$ is attached to the free end of the rope. A transverse pulse of wavelength $\lambda$ is produced at the lower end of the rope. The wavelength of the pulse, when it reaches the top of the rope, is $A\lambda$. Find $a^2$.

Answer 1

Tension at lower end,$T_1=mg$

Tension at upper end, $T_2=(m+m)g=2mg$

As we know, that

$velocity\propto \sqrt{T}$

$\therefore \lambda \propto \sqrt{T}$

Wavelength at lower end $=\lambda$

Wavelength when it reaches the top $=a\lambda$

Now,

$\frac{\lambda }{a\lambda }=\sqrt{\frac{T_1}{T_2}}$

$\Rightarrow \frac{1}{a}=\frac{1}{\sqrt{2}}$

$\Rightarrow a^2=2$