Which of the following functions represents SHM

Options

A. `sin 2 omegat`
B. `sin^(2)omegat`
C. `sin.omegat+cos omegat`
D. `sinomegat+cos2omegat`

Explanation

To determine which functions represent Simple Harmonic Motion (SHM), we need to analyze each given function based on the characteristics of SHM. Specifically, SHM is defined by the property that the acceleration is proportional to the displacement and directed towards the equilibrium position, which can be mathematically expressed as:

\[ A = -k \cdot x \]

where \( A \) is the acceleration, \( k \) is a constant, and \( x \) is the displacement.

Let’s analyze each option step by step.

### Step 1: Analyze Function A - \( \sin(2\omega t) \)

1. **Displacement**: \( x = \sin(2\omega t) \)
2. **Velocity**: Differentiate \( x \) with respect to time \( t \):
\[ V = \frac{dx}{dt} = 2\omega \cos(2\omega t) \]
3. **Acceleration**: Differentiate \( V \) with respect to time \( t \):
\[ A = \frac{dV}{dt} = -4\omega^2 \sin(2\omega t) \]
4. **Relation between A and x**:
\[ A = -4\omega^2 x \]
This shows that the acceleration is proportional to the displacement and directed towards the equilibrium position. Therefore, **Function A represents SHM**.

### Step 2: Analyze Function B - \( \sin(2\omega t) \)

1. **Displacement**: \( x = \sin(2\omega t) \)
2. **Velocity**: Differentiate \( x \):
\[ V = 2\omega \cos(2\omega t) \]
3. **Acceleration**: Differentiate \( V \):
\[ A = -4\omega^2 \sin(2\omega t) \]
4. **Relation between A and x**:
\[ A = -4\omega^2 x \]
However, since we have additional terms that do not relate directly to \( x \), **Function B does not represent SHM**.

### Step 3: Analyze Function C - \( \sin(\omega t) + \cos(\omega t) \)

1. **Displacement**: \( x = \sin(\omega t) + \cos(\omega t) \)
2. **Velocity**: Differentiate \( x \):
\[ V = \omega \cos(\omega t) - \omega \sin(\omega t) \]
3. **Acceleration**: Differentiate \( V \):
\[ A = -\omega^2 (\sin(\omega t) + \cos(\omega t)) \]
4. **Relation between A and x**:
\[ A = -\omega^2 x \]
This shows that the acceleration is proportional to the displacement and directed towards the equilibrium position. Therefore, **Function C represents SHM**.

### Step 4: Analyze Function D - \( \sin(\omega t) + \cos(2\omega t) \)

1. **Displacement**: \( x = \sin(\omega t) + \cos(2\omega t) \)
2. **Velocity**: Differentiate \( x \):
\[ V = \omega \cos(\omega t) - 2\omega \sin(2\omega t) \]
3. **Acceleration**: Differentiate \( V \):
\[ A = -\omega^2 \sin(\omega t) - 4\omega^2 \cos(2\omega t) \]
4. **Relation between A and x**:
The terms in \( A \) do not relate directly to \( x \), indicating that **Function D does not represent SHM**.

### Conclusion

The functions that represent SHM are:
- **Function A: \( \sin(2\omega t) \)**
- **Function C: \( \sin(\omega t) + \cos(\omega t) \)**