Question

The displacement $y$ of a particle, if given by $y=4 \cos ^{2}(t / 2) \sin (1000 t)$. This expression may be considered to be a result of the superposition of how many simple harmonic motions?

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (B)

3

Answer 2

$\because$ Equation of displacement of particle is
$y=4 \cos ^{2}\left(\frac{t}{2}\right) \sin (1000 t) \,...(i)$
or $y=4\left[\left(\frac{1+\cos t}{2}\right) \sin 1000 t\right]$
$\left(\because \cos ^{2} x=\frac{1+\cos 2 x}{2}\right)$
or $y=3 \sin 1000 t+2 \cos t \sin 1000 t$
or $y=2 \sin 1000 t+\sin 1001 t+\sin t 999 t$
[$2 \sin A \cos B=\sin (A+B)+\sin (A-B)$]
Hence, E (i) is the result of the superposition of three simple harmonic motion.