Question

Out of the following functions representing motion of a particle which represents SHM
(1) $y=\sin \omega t-\cos \omega t$
(2) $y=\sin ^{3} \omega t$
(3) $y=5 \cos \left(\frac{3 \pi}{4}-3 \omega t\right)$
(4) $y=1+\omega t+\omega^{2} t^{2}$

• A. Only (1) and (2)
• B. Only (1)
• C. Only (4) does not represent SHM
• D. Only (1) and (3)

Answer 1

Answer: (D)

$y=\sin \omega t-\cos \omega t$
$=\sqrt{2}\left[\frac{1}{\sqrt{2}} \sin \omega t-\frac{1}{\sqrt{2}} \cos \omega t\right]=\sqrt{2} \sin \left(\omega t-\frac{\pi}{4}\right)$
It represents a SHM with time period, $T=\frac{2 \pi}{\omega}$.
$y=\sin ^{3} \omega t=\frac{1}{4}[3 \sin \omega t-\sin 3 \omega t]$
It represents a periodic motion with time period $T=\frac{2 \pi}{\omega}$ but not SHM.
$y =5 \cos \left(\frac{3 \pi}{4}-3 \omega t\right)$
$=5 \cos \left(3 \omega t-\frac{3 \pi}{4}\right) [\because \cos (-\theta)=\cos \theta]$
It represents a SHM with time period,
$T=\frac{2 \pi}{3 \omega}$. $y=1+\omega t+\omega^{2} t^{2}$
It represents a non-periodic motion. Also it is not physically acceptable as the $y \rightarrow \infty$ as $t \rightarrow \infty$.