Out of the following functions representing motion of a particle which represents SHM?
1. `x=sin^(3)omegat`
2. `x=1+omegat+omega^(2)t^(2)`
3. `x=cosomegat+cos3omegat+cos5omegat`
4. `x=sinomegat+cosomegat`

Options

A. Only 1
B. Only 1 and 3
C. Only 1 and 4
D. Only 4

Explanation

To determine which of the given functions represents simple harmonic motion (SHM), we will analyze each function one by one.

### Step 1: Analyze the first function \( x = \sin^3(\omega t) \)

1. The function \( x = \sin^3(\omega t) \) can be rewritten using the identity for \( \sin^3 \theta \):
\[ \sin^3 \theta = \frac{3 \sin \theta - \sin(3\theta)}{4} \]
Applying this identity:
\[ x = \frac{3 \sin(\omega t) - \sin(3\omega t)}{4} \]
2. This function is periodic but not linear in terms of sine or cosine. Therefore, it does not represent SHM.

### Conclusion for Step 1:
- **Not SHM**.

### Step 2: Analyze the second function \( x = 1 + \omega t + \omega^2 t^2 \)

1. The function \( x = 1 + \omega t + \omega^2 t^2 \) is a quadratic function of time \( t \).
2. This function does not oscillate and is not periodic. It represents a parabolic motion.

### Conclusion for Step 2:
- **Not SHM**.

### Step 3: Analyze the third function \( x = \cos(\omega t) + \cos(3\omega t) + \cos(5\omega t) \)

1. The function \( x = \cos(\omega t) + \cos(3\omega t) + \cos(5\omega t) \) is a sum of cosines with different frequencies.
2. While this function is periodic, it does not represent a single frequency oscillation, which is a requirement for SHM.

### Conclusion for Step 3:
- **Not SHM**.

### Step 4: Analyze the fourth function \( x = \sin(\omega t) + \cos(\omega t) \)

1. The function \( x = \sin(\omega t) + \cos(\omega t) \) can be rewritten using the amplitude-phase form:
\[ x = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin(\omega t) + \frac{1}{\sqrt{2}} \cos(\omega t) \right) \]
This can be expressed as:
\[ x = \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]
2. This is a standard form of SHM, where the amplitude is \( \sqrt{2} \) and it oscillates with a frequency \( \omega \).

### Conclusion for Step 4:
- **Represents SHM**.

### Final Answer:
- The only function that represents simple harmonic motion is **Option 4: \( x = \sin(\omega t) + \cos(\omega t) \)**.

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