Two harmonic waves moving in the same direction superimpose to fo… - Vidyadip

MCQ

Two harmonic waves moving in the same direction superimpose to form a wave $\mathrm{x}=\mathrm{a} \cos (1.5 \mathrm{t}) \cos (50.5 \mathrm{t})$ where $t$ is in seconds. Find the period with which they beat (close to nearest integer)

A
6 s
B
4 s
C
1 s
D
2 s

✓

Answer

D. 2 s
The given equation can be written as
$x=\frac{a}{2} \cos [1.5+50.5] t+\frac{a}{2} \cos [50.5-1.5]$
$x=\frac{a}{2} \cos [52 t]+\frac{a}{2} \cos [49 t]$
Here, $2 \pi \mathrm{f}_{1}$ & $2 \pi \mathrm{f}_{2}=49$
$\mathrm{f}_{1}=\frac{52}{2 \pi}, \mathrm{f}_{2}=\frac{49}{2 \pi}$
$\therefore \mathrm{f}_{\text {Beat }}=\mathrm{f}_{1}-\mathrm{f}_{2}=\frac{3}{2 \pi} \mathrm{~Hz}$
$\therefore \mathrm{T}_{\text {Beat }}=\frac{1}{\mathrm{f}_{\text {Beat }}}=\frac{2 \pi}{3} \mathrm{sec}$
$=2.09 \mathrm{sec} \approx 2 \mathrm{sec}$

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