Question

The displacement $Y$ of a particle executing simple harmonic motion is given by

$Y=2\sin t{\cos }^2\frac{t}{2}$

This expression may be considered to be a result of the superposition of how many simple harmonic motions?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Given that
$Y=2\sin t{\cos }^2\frac{t}{2}$

$=2\sin t\cos \frac{t}{2}.\cos \frac{t}{2}$

=$(\sin \frac{3t}{2}+\sin \frac{t}{2})\cdot \cos \frac{t}{2}$

$\because 2\sin A\cos B=\sin (A+B)+\sin (A-B)$

$\therefore Y=\sin \frac{3t}{2}\cos \frac{t}{2}+\sin \frac{t}{2}\cos \frac{t}{2}$

$=\frac{1}{2}(\sin 2t+\sin t)+\frac{1}{2}\sin t$

$=\frac{1}{2}\sin 2t+\sin t$

So, $Y$ is a superposition of two $\text{SHM}$.

Hence, option $\left(\text{B}\right)$ is correct.

Why this question ? Trick: To find out number of Harmonic motion, decompose displacement equation into sinusoidal equations of motion.