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/arc correct?

• A. The time $T_{AO}=T_{OA}$
• B. The velocities of the two pulses (Pulse 1 and Pulse 2) are 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), (D)

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

Speed of the wave $v=\sqrt{\frac{T}{\mu }}$

where $\mu$ is mass of string per unit length and $T$. Since these are same for both the waves, both waves will have equal velocity.
Since length of the rope is also constant, time taken by the wave pulse will be same will travelling from O to A or A to O
$\therefore T_{AO}=T_{OA}$

At mid point, tension for both the waves is same, so speed is same. But direction will be different and hence velocities won't be same for both the pulse. Thus option B is incorrect.

Speed is higher at the point where tension is larger i.e. at A. Since frequency is constant, wavelength of the wave $\lambda =\frac{v}{\nu }$
$⟹\lambda \propto v$
When one moves from O to A , the tension becomes less and hence, $v$ decreases $\Rightarrow$ $\lambda$ become shorter

Velocity depends on tension, not frequency or wavelength. Hence D is also correct.