Question

The displacement of a particle in a periodic motion is given by $y=4{\cos }^2\left(\frac{t}{2}\right)\sin \left(1000t\right)$. This displacement may be considered as the result of superposition of $n$ independent harmonic oscillations. Here $n$ is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

The displacement equation is given as $y=4co{s}^2(\frac{t}{2})sin\left(1000t\right)$ .........(1)

Using $2co{s}^2\theta =1+cos2\theta$ in equation (1),

$y=2[1+cost]sin\left(1000t\right)$

$\therefore$ $y=2sin\left(1000t\right)+2sin\left(1000t\right)cost$ ............(2)

Using $2sinA$ $cosB=sin(A+B)+sin(A-B)$ in equation (2),

$y=2sin\left(1000t\right)+sin(1000t+t)+sin(1000t-t)$

$⟹$ $y=2sin\left(1000t\right)+sin\left(1001t\right)+sin\left(999t\right)$

Thus $3$ harmonic oscillations superimpose on each other to give the resultant motion $y$.