Question

Out of the following functions representing motion of a particle which represents SHM?

1. x = sin3 $\omega$t

2. x = 1 + $\omega$t + $\omega$2t2

3. x = cos $\omega$t + cos 3$\omega$t + cos 5$\omega$t

4. x = sin$\omega$t + cos$\omega$t

• A. Only 1
• B. Only 1 and 3
• C. Only 1 and 4
• D. Only 4

Answer 1

Answer: (D)

Only 4

Answer 2

(1) $x = \sin^{3}\omega t = \frac{1}{4}(3\sin\omega t - \sin 3\omega t)$

$(\because\sin 3\theta = 3\sin\theta - 4\sin^{3}\theta)$

It represent a periodic motion with time period

$T = \frac{2\pi}{\omega}$ but not SHM.

(2) $x = 1 + \omega t + \omega^{2}t^{2}$

It represents a non- periodic motion

(3) $x = \cos\omega t + \cos 3\omega t + \cos 5\omega t$

It represent a periodic motion with time period $T = \frac{2\pi}{\omega}$ but not SHM.

(4) $\sin\omega t + \cos\omega t = \sqrt{2}\left\lbrack \frac{1}{\sqrt{2}}\sin\omega t + \frac{1}{\sqrt{2}}\cos\omega t \right\rbrack$

$= \sqrt{2}\left\lbrack \sin\omega t\cos\frac{\pi}{4} + \sin\frac{\pi}{4}\cos\omega t \right\rbrack$

$= \sqrt{2}\sin(\omega t) + \frac{\pi}{4}$

It represents a SHM with time period $T = \frac{2\pi}{\omega}$