Step 1: Understanding the Concept:
A function is periodic if it repeats its values after a regular interval of time. The sum of multiple periodic functions is periodic if the ratio of their individual periods is a rational number.
Step 3: Detailed Explanation:
The function is \(y(t) = \sin \omega t + \cos 2\omega t + \sin 4\omega t\).
The individual periods are:
1. For \(\sin \omega t\): \(T_{1} = \frac{2\pi}{\omega}\).
2. For \(\cos 2\omega t\): \(T_{2} = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}\).
3. For \(\sin 4\omega t\): \(T_{3} = \frac{2\pi}{4\omega} = \frac{\pi}{2\omega}\).
The resultant period \(T\) is the Least Common Multiple (LCM) of \(T_{1}, T_{2}, \text{ and } T_{3}\).
\(T = \text{LCM} \left( \frac{2\pi}{\omega}, \frac{\pi}{\omega}, \frac{\pi}{2\omega} \right) = \frac{2\pi}{\omega}\).
Since a single common period exists, the overall motion is periodic. It is not SHM because it consists of multiple frequencies.
Step 4: Final Answer:
The motion represented by the function is periodic.