Question

A uniform rope of length $10\text{m}$ and mass $15\text{kg}$ hangs vertically from a rigid support. A block of mass $5\text{kg}$ is attached to the free end of the rope. A transverse pulse of wavelength $0.08\text{m}$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope will be

• A. $0.08\text{m}$
• B. $0.04\text{m}$
• C. $0.16\text{m}$
• D. $0\text{m}$

Answer 1

The correct option is C $0.16\text{m}$

Wavelength at lower end is $0.08\text{m}$
Velocity of wave on string $=\sqrt{\frac{T}{\mu }}$
$\mu =\frac{15}{10}=1.5$
Tension at lower end $\Rightarrow T=50\text{N}$
Tension at upper end $\Rightarrow T=200\text{N}$
$v_{\text{lower}}=\sqrt{\frac{50}{1.5}}$
$v_{\text{upper}}=\sqrt{\frac{200}{1.5}}$
frequency is constant hence $\frac{v_1}{{\lambda }_1}=\frac{v_2}{{\lambda }_2}$
$\frac{\sqrt{50}}{0.08}=\frac{\sqrt{200}}{\lambda }$
$\Rightarrow \lambda =2\times 0.08$
$=0.16\text{m}$