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 step by step.

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

1. The function \( x = \sin^3(\omega t) \) can be rewritten using the trigonometric identity:
\[ \sin^3(\theta) = \frac{3\sin(\theta) - \sin(3\theta)}{4} \]
Thus, we have:
\[ x = \frac{3\sin(\omega t) - \sin(3\omega t)}{4} \]
2. This expression is a combination of two oscillatory functions (one with frequency \( \omega \) and the other with frequency \( 3\omega \)). Therefore, it does not represent simple harmonic motion (SHM) because SHM requires a single frequency.

**Conclusion for Step 1:** This function does not represent 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. As \( t \) increases, this function does not repeat its values; it continuously increases without oscillation.
3. Since SHM requires periodic motion, and this function is not periodic, it cannot represent SHM.

**Conclusion for Step 2:** This function does not represent 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 cosine functions with different frequencies.
2. While each cosine function is periodic, the combination of multiple frequencies results in a more complex periodic motion, which is not simple harmonic motion.
3. Therefore, this function does not represent SHM.

**Conclusion for Step 3:** This function does not represent SHM.

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

1. We can rewrite this function using the amplitude-phase form:
\[ x = \sin(\omega t) + \cos(\omega t) = \sqrt{2} \left(\frac{1}{\sqrt{2}} \sin(\omega t) + \frac{1}{\sqrt{2}} \cos(\omega t)\right) \]
2. Recognizing that \( \frac{1}{\sqrt{2}} = \cos(45^\circ) \) and \( \frac{1}{\sqrt{2}} = \sin(45^\circ) \), we can express this as:
\[ x = \sqrt{2} \sin\left(\omega t + 45^\circ\right) \]
3. This is in the standard form of SHM, \( x = A \sin(\omega t + \phi) \), where \( A = \sqrt{2} \) and \( \phi = 45^\circ \).

**Conclusion for Step 4:** This function represents SHM.

### Final Conclusion

Out of the given functions, only the fourth function \( x = \sin(\omega t) + \cos(\omega t) \) represents simple harmonic motion (SHM).

### Summary of Results
- **Function 1:** Not SHM
- **Function 2:** Not SHM
- **Function 3:** Not SHM
- **Function 4:** Represents SHM