Question

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at $O$. A transverse wave pulse (Pulse 1) of wavelength ${\lambda }_0$ is produced at point $O$ on the rope. The pulse takes time $T_{OA}$ to reach point $A$. If the wave pulse of wavelength ${\lambda }_0$ is produced at point $A$ (Pulse 2) without disturbing the position of $M$ it takes time $T_{AO}$ to reach point $O$. Which of the following options is/are correct?

• A. The time $T_{AO}=T_{OA}$
• B. The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope
• C. The wavelength of Pulse 1 becomes longer when it reaches point $A$
• D. The velocity of any pulse along the rope is independent of its frequency and wavelength

Answer 1

Answer: (A), (B), (D)

$(A)V=\sqrt{\frac{T}{\mu }}$
Tension at midpoint
$T=\left(M+\frac{\mu L}{2}\right)g$
Velocity at midpoint
$V_1=\sqrt{\frac{\left(M+\frac{\mu L}{2}\right)g}{\mu }}$
Velocity at midpoint is same for both pulses
(B) As pulse goes down, Tension decreases, V also decreases hence $\lambda$ decreases
(D) At any height $x$
$V=\sqrt{\frac{(M+\mu x)g}{\mu }}=\frac{dx}{dt}$
$\underset{0}{\overset{t}{\int }}dt=\sqrt{\frac{\mu }{g}}{\int }_0^x\frac{dx}{\sqrt{M+\mu x}}$
$T_{AO}=T_O$