Question

Which of the following functions represent a simple harmonic motion?

• A. $\sin \omega t-\cos \omega t$
• B. ${\sin }^2\omega t$
• C. $\sin \omega t+{\sin }^2\omega t$
• D. $\sin \omega t-{\sin }^2\omega t$

Answer 1

The correct option is A $\sin \omega t-\cos \omega t$

A function represents a simple harmonic motion if and only if $\frac{d^{2}y}{d{t}^2}\propto -y$

For function $y=sinwt-coswt$

Differentiating w.r.t time gives $\frac{dy}{dt}=wcoswt+wsinwt$

Differentiating again w.r.t time gives $\frac{d^{2}y}{d{t}^2}=-w^{2}sinwt+w^{2}coswt$

$\frac{d^{2}y}{d{t}^2}=-w^2[sinwt-coswt]$

$⟹\frac{d^{2}y}{d{t}^2}=-w^{2}y$

Hence it represents SHM.