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. $\lambda \sqrt{\frac{M-m}{m}}$
• B. $\lambda \frac{M-m}{m}$
• C. $\lambda \sqrt{\frac{m}{M+m}}$
• D. $\lambda \sqrt{\frac{M+m}{m}}$

Answer 1

The correct option is D $\lambda \sqrt{\frac{M+m}{m}}$

the frequency at end and at the begining will remain same.
if one wave is generated at bottom in one sec, only one will reach the top.
hence we need equate frequency at top and bottom
$\frac{v_1}{{\lambda }_1}=\frac{v_2}{{\lambda }_2}$
$v_2\alpha \sqrt{tensionofrope}\alpha \sqrt{m+M}$
$v_1\alpha \sqrt{tensionofrope}\alpha \sqrt{m}$

$⟹\frac{{\lambda }_2}{{\lambda }_1}=\frac{v_2}{v_1}=\sqrt{\frac{M+m}{m}}$

$⟹{\lambda }_2=\lambda \sqrt{\frac{M+m}{m}}$