Question

Which of the following equation does not represent a SHM?

• A. $cos\omega t + sin\omega t$
• B. $sin\omega t-cos\omega t$
• C. $1 - sin2\omega t$
• D. $sin\omega t+cos\left(\omega t +\alpha\right)$

Answer 1

Answer: (C)

$1 - sin2\omega t$

Answer 2

(a) $\left(cos \omega\, t + sin \omega \,t\right)$ is a periodic function. It can also be written as $=\frac{\sqrt{2}}{\sqrt{2}}sin \,\omega t +\frac{\sqrt{2}}{\sqrt{2}}=cos\,\omega t$ $=\sqrt{2}\,\left(cos \frac{\pi}{4} sin \,\omega t +sin \frac{\pi}{4} cos \,\omega t\right)$ $=\sqrt{2}sin\, \left(\omega t+ \frac{\pi}{4}\right)=\sqrt{2}\,sin \left(\omega t+\frac{\pi}{4}+2\pi\div\right)$ $=\sqrt{2}\,sin\left[\omega\left(t+\frac{2\pi}{\omega} \right)+\frac{\pi}{4}\right]$ This represent a simple harmonic function with period $\frac{2\pi}{\omega}$ and phase $\frac{\pi}{4}.$ (b) $sin\omega t - cos\omega$ t is a periodic function. It can be written as $=\sqrt{2}\left[sin \omega t cos \frac{\pi}{4}-cos \omega t sin \frac{\pi}{4} \right]$ $=\sqrt{2}sin \left(\omega t-\frac{\pi}{4}\div \right)=\sqrt{2}sin \left[\omega\left(t+\frac{2\pi}{\omega}\div\right)-\frac{\pi}{4}\right]$ This represent a simple harmonic function with period $\frac{2\pi}{\omega}.$ (c) $F\left(t\right) = 1 - sin 2\omega t$ This is a non periodic function. (d) $F\left(t\right) = sin\omega t + cos\left(\omega t + \alpha\right)$ also represent a simple harmonic function.