Question

A function of time is represented as follows \( \sin \omega t + \cos 2\omega t + \sin 4\omega t \). The motion represented by it is:

• A. non-periodic
• B. periodic
• C. both non-periodic and periodic
• D. data insufficient

Hint:

If frequencies are integer multiples → function is always periodic!

Answer 1

The Correct Option is B

Concept: Sum of periodic functions is periodic if their time periods are commensurable.

Step 1: Individual time periods: \[ T_1 = \frac{2\pi}{\omega}, \quad T_2 = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}, \quad T_3 = \frac{2\pi}{4\omega} = \frac{\pi}{2\omega} \]

Step 2: Ratio: \[ T_1 : T_2 : T_3 = 4 : 2 : 1 \] (All are multiples of smallest period)

Step 3: Hence, common time period exists. \[ \therefore \text{motion is periodic} \]