A composition string is made up by joining two strings of differe… - Vidyadip

MCQ

A composition string is made up by joining two strings of different masses per unit length $\rightarrow \mu$ and $4\mu$ . The composite string is under the same tension. A transverse wave pulse $: Y = (6 mm) \,\,sin\,\,(5t + 40x),$ where $‘t’$ is in seconds and $‘x’$ in meters, is sent along the lighter string towards the joint, the percentage of power transmitted to the heavier string through the joint is approximately ..... $\%$

A
$33$
✓
$89$
C
$67$
D
$75$

✓

Answer

Correct option: B.

$89$

b
Power, $P=\frac{1}{2} \mu \omega^{2} A^{2} v$

Amplitude of transmitted wave, $A_{t}=\frac{2 Z_{1}}{Z_{1}+Z_{2}} A=\frac{2 \sqrt{\mu T}}{\sqrt{\mu T}+\sqrt{4 \mu T}} A=\frac{2}{3} A$

Thus, $P=\frac{1}{2} \mu \omega^{2} A^{2} v$

$=\frac{1}{2} \times 4 \mu \times \omega^{2} \times \frac{4 A^{2}}{9} \times \sqrt{\frac{T}{4 \mu}}=\left(\frac{1}{2} \times \mu \times \omega^{2} \times A^{2} \times \sqrt{\frac{T}{\mu}}\right) \times \frac{8}{9}$

$=\frac{8}{9} \times$ Initial power

Thus power transmitted $=\frac{8}{9} \times 100=89 \%$

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