Out of the following functions representing motion of a particle which represents SHM
I. `y = sin omega t - cos omega t`
II. `y = sin^(3)omega t`
III. `y = 5 cos ((3 pi)/(4)-3 omega t)`
IV. `y = 1 + omega t + omega^(2)t^(2)`

Options

A. Only (IV) does not represent SHM
B. (I) and (II)
C. (I) and (III)
D. Only (I)

Answer

To determine which of the given functions represents Simple Harmonic Motion (SHM), we will analyze each function based on the characteristics of SHM.

### Step-by-Step Solution:

1. **Understanding SHM**:
- SHM is characterized by a restoring force that is proportional to the displacement from the mean position and acts in the opposite direction. Mathematically, this can be expressed as \( a = -\omega^2 y \), where \( a \) is acceleration, \( \omega \) is angular frequency, and \( y \) is displacement.

2. **Analyzing Each Function**:
- **Function I: \( y = \sin(\omega t) - \cos(\omega t) \)**
- This function is a combination of sine and cosine functions. Since both sine and cosine functions represent SHM when their powers are 1 and they have the same angular frequency, this function can represent SHM.

- **Function II: \( y = \sin^3(\omega t) \)**
- Here, the sine function is raised to the power of 3. Since SHM requires the power of sine or cosine to be 1, this function does not represent SHM.

- **Function III: \( y = 5 \cos\left(\frac{3\pi}{4} - 3\omega t\right) \)**
- This function is a cosine function with a coefficient (5) and a phase shift. The angular frequency is \( 3\omega \), but since it is still a cosine function with power 1, it represents SHM.

- **Function IV: \( y = 1 + \omega t + \omega^2 t^2 \)**
- This is a polynomial function (quadratic) in \( t \). Polynomial functions do not represent SHM because they do not oscillate about a mean position. Thus, this function does not represent SHM.

3. **Conclusion**:
- The functions that represent SHM are:
- Function I: \( y = \sin(\omega t) - \cos(\omega t) \)
- Function III: \( y = 5 \cos\left(\frac{3\pi}{4} - 3\omega t\right) \)
- Therefore, the correct options that represent SHM are I and III.

### Final Answer:
The functions that represent SHM are I and III.

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