First slide

Travelling wave

Question

A heavy but uniform rope of length L is suspended from a ceiling. A particle is dropped from the ceiling at the instant when the bottom end is given a transverse wave pulse. Where will the particle meet the pulse?

Moderate

A

at a distance $\frac{2 L}{3}$ from the bottom
B

at a distance $\frac{L}{3}$ from the bottom
C

at distance $\frac{3 L}{4}$ from the bottom
D

at a distance $\frac{L}{4}$ from the bottom

Solution

$\begin{array}{l} [Step 1] : V = \sqrt{g h} ∴ \frac{d h}{d t} = \sqrt{g h} h i s t h e h e i g h t t r a v e l l e d b y t h e w a v e f r o m t h e b o t t o m o f t h e r o p e . \\ [Step 2] :∴ t = \int_{0}^{h} \frac{d h}{\sqrt{g h}} or t = 2 \sqrt{\frac{h}{g}} \end{array}$
[Concept]: Now at the time of meeting
[Step 3]: Time of fall of particle = time of wave pulse on reaching up to these
[Step 4]: $\sqrt{\frac{2 (L - h)}{g}} = 2 \sqrt{\frac{h}{g}}$ where L-h is the distance through which the body dropped falls.
[Step 5]: $∴ h = \frac{L}{3}$

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