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 } $
$\int\limits_{0}^{ t } d t=\sqrt{\frac{\mu}{ g }} \int_{0}^{ x } \frac{ dx }{\sqrt{ M +\mu x }} $
$ T _{ A O}= T _{ O }$