Identify the function which represent a periodic motion.

Options

A. `log_(e)(omegat)`
B. `sinomegat+cosomegat`
C. `e^(-omegat)`
D. `e^(omegat)`

Explanation

To identify the function that represents periodic motion, we need to understand what periodic motion is. Periodic motion is defined as motion that repeats itself after a fixed period of time. Let's analyze each option provided in the question to determine which one represents periodic motion.

### Step-by-Step Solution:

1. **Understand Periodic Motion**:
- Periodic motion is characterized by a function that repeats its values in regular intervals of time.

2. **Evaluate the First Option: log(ωt)**:
- The function log(ωt) is a logarithmic function.
- Logarithmic functions are continuously increasing and do not repeat values.
- Therefore, this function does not represent periodic motion.
- **Conclusion**: This option is incorrect.

3. **Evaluate the Second Option: sin(ωt) + cos(ωt)**:
- Both sine and cosine functions are periodic, with a period of 2π.
- The sum of two periodic functions with the same period is also periodic.
- Therefore, sin(ωt) + cos(ωt) represents a periodic motion with a period of 2π.
- **Conclusion**: This option is correct.

4. **Evaluate the Third Option: e^(-ωt)**:
- The function e^(-ωt) is an exponential decay function.
- Exponential functions do not repeat values; they either grow or decay continuously.
- Therefore, this function does not represent periodic motion.
- **Conclusion**: This option is incorrect.

5. **Evaluate the Fourth Option: e^(ωt)**:
- Similar to the previous option, e^(ωt) is an exponential growth function.
- Like the decay function, it does not repeat values and is not periodic.
- Therefore, this function does not represent periodic motion.
- **Conclusion**: This option is incorrect.

### Final Answer:
The only function that represents periodic motion among the given options is **sin(ωt) + cos(ωt)**.