Wave Motion 1 Question 9
9. 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 $T _{O A}$ 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 $T _{A O}$ to reach point $O$. Which of the following options is/are correct?
(a) The time $T _{A O}=T _{O A}$
(2017 Adv.)
(b) The wavelength of Pulse 1 becomes longer when it reaches point $A$
(c) The velocity of any pulse along the rope is independent of its frequency and wavelength
(d) The velocities of the two pulses (Pulse 1 and Pulse 2) are the same at the mid-point of rope
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Answer:
Correct Answer: 9. $(a, c, d)$
Solution:
- $v=\sqrt{\frac{T}{\mu}}$, so speed at any position will be same for both pulses, therefore time taken by both pulses will be same.
$$ \begin{aligned} & \lambda f & =v \\ \Rightarrow & & \lambda=\frac{v}{f} \\ \Rightarrow & & \lambda \propto v \propto T \end{aligned} $$
since when pulse 1 reaches at $A$ tension and hence speed decreases therefore $\lambda$ decreases.
NOTE
If we refer velocity by magnitude only, then option $(a, c, d)$ will be correct, else only $(a, c)$ will be correct.
