Question

A uniform rope of length $12\text{m}$ and mass $6\text{kg}$ hangs vertically from a rigid support. A block of mass $2\text{kg}$ is attached to the free end of the rope. A transverse pulse of wavelength $0.06\text{m}$ is produced at the lower end of the rope. What is the wavelength of the pulse when it reaches the top of the rope ?

• A. $0.06\text{m}$
• B. $0.12\text{m}$
• C. $0.09\text{m}$
• D. None of these

Answer 1

The correct option is B $0.12\text{m}$

Velocity of wave travelling in the rope is given by
$v=\sqrt{\frac{T}{\mu }}$
$\mu$ is constant as rope is uniform.

$\Rightarrow \frac{v_{\text{top}}}{v_{\text{bottom}}}=\sqrt{\frac{T_{top}}{T_{bottom}}}$


Tension at the top, $T_{top}=(2+6)g$
& $T_{bottom}=2g$

$\Rightarrow \frac{v_{\text{top}}}{v_{\text{bottom}}}=\sqrt{\frac{8g}{2g}}=4$
$\Rightarrow \frac{f\times {\lambda }_{\text{top}}}{f\times {\lambda }_{\text{bottom}}}=2$
as frequency does not change.

$\therefore {\lambda }_{\text{top}}={\lambda }_{\text{bottom}}\times 2=0.12\text{m}$