Question

A uniform rope of length $L$ and mass $m_1$ hangs vertically from a rigid support. A block of mass $m_2$ is attached to the free end of the rope. A transverse pulse of wavelength ${\lambda }_1$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is ${\lambda }_2$. The ratio ${\lambda }_2/{\lambda }_1$ is:

• A. $\sqrt{\frac{m_1}{m_2}}$
• B. $\sqrt{\frac{m_1+m_2}{m_2}}$
• C. $\sqrt{\frac{m_2}{m_1}}$
• D. $\sqrt{\frac{m_1+m_2}{m_1}}$

Answer 1

The correct option is B $\sqrt{\frac{m_1+m_2}{m_2}}$

$T_1=m_{2}g$

$T_2=(m_1+m_2)g$

Velocity $\propto \sqrt{T}$

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

$\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{\sqrt{T_1}}{\sqrt{T_2}}$

$\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\sqrt{\frac{m_1+m_2}{m_2}}$