The displacement of a particle executing periodic motion is given by y - 4 cos2 t2 sin 1000 t- The number of independent constituent simple harmonic motion is

Given y = 4 cos^2 ( t2 ) sin (1000 t) Using 2 cos^2 θ = 1+ cos 2 θ, we get y = 2 × 2 cos^2 ( t2 ) sin (1000 t) y = 2 × (1 + cos t) sin (1000 t) y = 2 sin (100 ...

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Standard XII

Physics

Amplitude Variation

The displacem...

Question

The displacement of a particle executing periodic motion is given by $y = 4 {cos}^{2} (\frac{t}{2}) sin (1000 t)$ . The number of independent constituent simple harmonic motion is

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Solution

The correct option is C $3$
Given
$y = 4 {cos}^{2} (\frac{t}{2}) sin (1000 t)$
Using $2 {cos}^{2} θ = 1 + cos 2 θ$ , we get
$y = 2 \times 2 {cos}^{2} (\frac{t}{2}) sin (1000 t)$
$y = 2 \times (1 + cos t) sin (1000 t)$
$y = 2 sin (1000 t) + 2 sin (1000 t) cos t$
$y = 2 sin (1000 t) + sin (1000 t + t) + sin (1000 t - t)$
[Using $2 sin A cos B = sin (A + B) + sin (A - B)$ ]
$y = 2 sin (1000 t) + sin (1001 t) + sin 999 t$
$y = sin (1000 t) + sin (1000 t) + sin (1001 t) + sin (999 t)$
Thus, the given expression is composed of four SHMs, out of which two are the same. This means that the given expression is the result of $3$ independent harmonics.

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The displacement of a particle executing periodic motion is given by y=4cos2(t2)sin(1000t). The number of independent constituent simple harmonic motion is

The displacement of a particle executing periodic motion is given by $y = 4 {cos}^{2} (\frac{t}{2}) sin (1000 t)$ . The number of independent constituent simple harmonic motion is