Identify the function which represents a non-periodic motion?

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1.  **Definition of Periodic Motion:** A function $f(t)$ represents periodic motion if it repeats its value at regular intervals of time, i.e., $f(t) = f(t + T)$, where $T$ is the time period. Trigonometric functions like sine and cosine are fundamental periodic functions.
2.  **Analysis of Options:**
    *   $\sin \omega t$: This is a basic periodic function with period $T = 2\pi/\omega$.
    *   $\sin \omega t + \cos \omega t$: This represents the superposition of two periodic functions with the same frequency. It can be rewritten as $\sqrt{2}\sin(\omega t + \pi/4)$, which is a Simple Harmonic Motion (a type of periodic motion) with period $T = 2\pi/\omega$.
    *   $\sin(\omega t + \pi/4)$: This is a sine function with a phase shift, which remains periodic with period $T = 2\pi/\omega$.
    *   $e^{-\omega t}$: This is an exponential function. It represents exponential decay where the value decreases continuously towards zero as time increases. It never repeats its value. Therefore, it represents **non-periodic motion**.
3.  **Conclusion:** The function $e^{-\omega t}$ is the only non-periodic function among the options.

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