Identify function which represent periodic motion

Options

A. `e^(omega t)`
B. `log_e(omega t)`
C. ` sin (omega t) + cos (omega t)`
D. `e^(-omega t)`

Answer

To identify a function that represents periodic motion, we need to analyze the characteristics of various functions. Periodic motion is defined as motion that repeats itself after a fixed interval of time, known as the period. Here’s a step-by-step solution to identify such functions:

### Step 1: Understand the Definition of Periodic Motion
Periodic motion occurs when a function repeats its values at regular intervals. Mathematically, a function \( f(t) \) is periodic if there exists a positive constant \( T \) (the period) such that:
\[ f(t + T) = f(t) \]
for all \( t \).

### Step 2: Analyze the Sine Function
The sine function is a classic example of a periodic function. The sine function can be expressed as:
\[ f(t) = \sin(\omega t) \]
where \( \omega \) is the angular frequency. The sine function has a period of \( 2\pi \):
\[ \sin(\omega(t + T)) = \sin(\omega t + \omega T) \]
Setting \( \omega T = 2\pi \), we find that \( T = \frac{2\pi}{\omega} \). Therefore, the sine function is periodic.

### Step 3: Analyze the Exponential Function
Next, consider the exponential function:
\[ f(t) = e^{\omega t} \]
As \( t \) increases, \( e^{\omega t} \) continuously increases and does not repeat its values. Therefore, it is not periodic.

### Step 4: Analyze the Logarithmic Function
Now, let’s look at the logarithmic function:
\[ f(t) = \log(\omega t) \]
The logarithmic function also continuously increases as \( t \) increases and does not repeat its values. Hence, it is not periodic.

### Step 5: Analyze the Combination of Sine and Cosine Functions
Finally, consider the function:
\[ f(t) = \sin(\omega t) + \cos(\omega t) \]
Both sine and cosine functions are periodic with a period of \( 2\pi \). The sum of two periodic functions with the same period is also periodic. Therefore, this function is periodic.

### Conclusion
From the analysis, the functions that represent periodic motion are:
1. \( \sin(\omega t) \)
2. \( \cos(\omega t) \)
3. \( \sin(\omega t) + \cos(\omega t) \)

The exponential function \( e^{\omega t} \) and the logarithmic function \( \log(\omega t) \) do not represent periodic motion.