# The function of time representing a simple harmonic motion

Question:

The function of time representing a simple harmonic motion with a period of $\frac{\pi}{\omega}$ is :

1. (1) $\sin (\omega t)+\cos (\omega t)$

2. (2) $\cos (\omega t)+\cos (2 \omega t)+\cos (3 \omega t)$

3. (3) $\sin ^{2}(\omega t)$

4. (4) $3 \cos \left(\frac{\pi}{4}-2 \omega t\right)$

Correct Option: , 4

Solution:

(4)

Time period $\mathrm{T}=\frac{2 \pi}{\omega^{\prime}}$

$\frac{\pi}{\omega}=\frac{2 \pi}{\omega^{\prime}}$

$\omega^{\prime}=2 \omega \rightarrow$ Angular frequency of SHM

Option (c)

$\sin ^{2} \omega t=\frac{1}{2}\left(2 \sin ^{2} \omega t\right)=\frac{1}{2}(1-\cos 2 \omega t)$

Angular frequency of $\left(\frac{1}{2}-\frac{1}{2} \cos 2 \omega t\right)$ is $2 \omega$

Option (d) Angular frequency of SHM

$3 \cos \left(\frac{\pi}{4}-2 \omega t\right)$ is $2 \omega$.

So option (c) & (d) both have angular frequency $2 \omega$ but option (d) is direct answer.