Which of the following functions of time represent periodic motion. For theones which are periodic, find the time period.
(i) $\cos 2 ω t + \sin 5 ω t$ (ii) $\sin^{2} 2 ω t$

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$Only (ii) is periodic, \frac{π}{4 ω}$

$Only (i) is periodic, \frac{π}{10 ω}$

$Both are periodic, \frac{π}{2 ω}, \frac{2 π}{ω}$

$Both are periodic, \frac{2 π}{ω}, \frac{π}{2 ω}$

answer is A.

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Detailed Solution

$\cos 2 ω t + \sin 5 ω t$
As per Fourier theorem, $\cos 2 ω t + \sin 5 ω t$ is a periodic function.
Time period of $\cos 2 ω t$ is $T_{1} = \frac{2 π}{2 ω}$ . Time period of $\sin 5 ω t$ is $T_{2} = \frac{2 π}{5 ω}$
Its time period is the LCM of $T_{1}$ and $T_{2}$ , which is $\frac{2 π}{ω}$ .
$\sin^{2} 2 ω t = \frac{1}{2} - \frac{\cos 4 ω t}{2}$
It is a periodic function with time period = $\frac{2 π}{4 ω} = \frac{π}{2 ω}$