The displacement y of a particle executing periodic motion is given by $y = 4 {cos}^{2} (t / 2) sin (1000 t)$ . This expression may be considered to be a result of the superposition of

two SHMs

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three SHMs

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four SHMs

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five SHMs

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Solution

The correct option is B three SHMs
$y = 4 c o s^{2} (t / 2) s i n 1000 t = 2 c o s (t / 2) (2 c o s (t / 2) s i n 1000 t)$ $\Rightarrow y = 2 c o s (t / 2) (s i n (1000 t + t / 2) + s i n (1000 t - t / 2))$ $\Rightarrow y = 2 c o s (t / 2) s i n (999.5 t) + 2 c o s (t / 2) s i n 1000.5 t$ $\Rightarrow y = s i n (999.5 t + 0.5 t) + s i n (999.5 t - 0.5 t) + s i n (1000.5 t + 0.5 t) + s i n (1000.5 t - 0.5 t)$ $\Rightarrow y = 2 s i n (1000 t) + s i n (999 t) + s i n (1001 t)$ From $y$ is sum of three sinusoids i.e., three SHMs.

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y

y = 4 {cos}^{2} (\frac{t}{2}) sin (1000 t)

y = 4 {cos}^{2} (\frac{t}{2}) sin 1000 ω t .

y = 4 c o s^{2} (\frac{t}{2}) s i n (1000 t)

y = 4 {cos}^{2} (t / 2) sin (100 t)

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y = 4 {cos}^{2} (\frac{1}{2} t) sin (1000 t)

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