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 is produced at point O on the rope. The pu

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 $λ_{0}$ is produced at point O on the rope. The pulse takes time $T_{O A}$ to reach point A. If the wave pulse of wavelength $λ_{0}$ is produced at point A (Pulse 2) without disturbing the position of M, it takes time $T_{A O}$ to reach point O. Which of the following options is/are correct [2017]

Smart Achievers · Accepted Answer

(1, 4)

$Wavelength of pulse, λ = \frac{v}{f} = \frac{1}{f} \sqrt{\frac{T}{μ}} or, T \propto \sqrt{T}$

Where $T =$ tension of string.

Here $T_{1} > T_{2} ∴ λ_{1} > λ_{2}$

The velocities of the two pulses cannot be same at mid-point as velocity being vector quantity has direction.

$V = \sqrt{\frac{T}{μ}},$ so speed at any position will be same for both pulses, therefore time taken by both pulses will be same i.e., $T_{A O} = T_{O A}$

Solution

1 The time $T_{A O} = T_{O A}$

2 The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the midpoint of rope

3 The wavelength of Pulse 1 becomes longer when it reaches point A

4 The velocity of any pulse along the rope is independent of its frequency and wavelength

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