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 $y\to \infty$ as $t\to \infty$.