JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Elasticity & SHM

A period of $\frac{\pi }{\omega }$ means an angular frequency $\Omega =\frac{2\pi }{T}=2\omega$, and simple harmonic motion requires a single sinusoid of that frequency about a fixed mean. $\sin \omega t+\cos \omega t=\sqrt{2}\sin \left(\omega t+\frac{\pi }{4}\right)$ oscillates at $\omega$, and a sum of cosines at $\omega$, $2\omega$ and $3\omega$ is not a single sinusoid at all. ${\sin }^2\omega t=\frac{1-\cos 2\omega t}{2}$ does repeat after $\frac{\pi }{\omega }$, but it swings about $\frac{1}{2}$ rather than about zero, so the function itself is not proportional to its own second derivative. $3\cos \left(\frac{\pi }{4}-2\omega t\right)$ is a single sinusoid of angular frequency $2\omega$ about zero, so it is the simple harmonic motion of period $\frac{\pi }{\omega }$.