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)

(a) $\left(cos\omega t+sin\omega t\right)$ is a periodic function. It can also be written as
$=\frac{\sqrt{2}}{\sqrt{2}}sin\omega t+\frac{\sqrt{2}}{\sqrt{2}}=cos\omega t$
$=\sqrt{2}\left(cos\frac{\pi }{4}sin\omega t+sin\frac{\pi }{4}cos\omega t\right)$
$=\sqrt{2}sin\left(\omega t+\frac{\pi }{4}\right)=\sqrt{2}sin\left(\omega t+\frac{\pi }{4}+2\pi ÷\right)$
$=\sqrt{2}sin\left[\omega \left(t+\frac{2\pi }{\omega }\right)+\frac{\pi }{4}\right]$
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
$=\sqrt{2}\left[sin\omega tcos\frac{\pi }{4}-cos\omega tsin\frac{\pi }{4}\right]$
$=\sqrt{2}sin\left(\omega t-\frac{\pi }{4}÷\right)=\sqrt{2}sin\left[\omega \left(t+\frac{2\pi }{\omega }÷\right)-\frac{\pi }{4}\right]$
This represent a simple harmonic function with period $\frac{2\pi }{\omega }.$
(c) $F\left(t\right)=1-sin2\omega t$
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.