Question

Identify the function which represents a periodic motion

• A. $e^{\omega t}$
• B. $log_{e} \left(\omega t\right)$
• C. $sin\,\omega t+cos\,\omega t$
• D. $e^{-\omega t}$

Answer 1

The Correct Option is C

To identify which function represents a periodic motion, we need to understand the concept of periodic functions in mathematics and physics. A periodic motion is one that repeats itself after a regular interval of time. Mathematically, a function \( f(t) \) is said to be periodic with period \( T \) if:

\(f(t + T) = f(t)\) for all values of \( t \).

Let's evaluate each option:

1. \(e^{\omega t}\): This is an exponential function. Exponential functions increase or decrease continuously and do not repeat their values over regular intervals. Therefore, this function is not periodic.
2. \(log_{e}(\omega t)\): The natural logarithmic function also does not repeat itself at regular intervals. Logarithmic functions are defined for positive values and grow unbounded as the argument increases. Hence, this function is not periodic.
3. \(sin\,\omega t+cos\,\omega t\): Both sine and cosine functions exhibit periodic behavior with a fundamental period of \(2\pi\). The combination \(sin\,\omega t + cos\,\omega t\) is periodic with the same period \(\frac{2\pi}{\omega}\), provided that \(\omega\) is a constant. Thus, this function represents periodic motion.
4. \(e^{-\omega t}\): Similar to the first option, this is an exponentially decaying function. It continuously decreases and will not repeat at regular intervals, hence it is not periodic.

Based on the above analysis, option 3 (\(sin\,\omega t + cos\,\omega t\)) is the correct answer as it represents periodic motion. Both sine and cosine are fundamental periodic functions, making their combination periodic as well.