Physics - SHM Question with Solution | TestHub

Let us analyze each option to determine if it represents oscillatory motion.

For the equation $y=A\sin \omega t+A\cos 2\omega t$, the motion is a superposition of two harmonic functions with commensurable frequencies. This represents a periodic and oscillatory motion.

For the equation $x=A\cos \left(\omega t-\frac{\pi }{4}\right)$, the motion is simple harmonic, which is inherently oscillatory.

For the equation $y=A\sin \omega t+A\cos (\omega t+\pi /2)$, we can simplify it using the trigonometric identity $\cos (\theta +\pi /2)=-\sin \theta$:
$y=A\sin \omega t-A\sin \omega t=0$
Since the displacement $y$ is zero for all time $t$, the particle remains at rest at the origin. Therefore, this does not represent oscillatory motion.

For the equation $y=A\sin \left(\omega t-\frac{\pi }{4}\right)$, the motion is simple harmonic, which is oscillatory.

Thus, the equation that does not represent oscillatory motion is $y=A\sin \omega t+A\cos (\omega t+\pi /2)$.

Answer: $y=A\sin \omega t+A\cos (\omega t+\pi /2)$