Which of the following functionss represents a simple harmonic oscillation ?

Options

A. `sin omega t- cos omega t`
B. `sin^(2) omegat`
C. `sin omegat+sin2 omega t`
D. `sin omega t- sin 2 omega t`

Explanation

To determine which of the given functions represents simple harmonic motion (SHM), we need to analyze the conditions for SHM. The key condition is that the acceleration of the oscillating particle must be directly proportional to the negative of its displacement from the equilibrium position.

### Step-by-Step Solution:

1. **Understand the Condition for SHM**:
- For a function to represent SHM, the acceleration \( a \) must satisfy the condition:
\[ a = -k \cdot y \]
where \( k \) is a positive constant and \( y \) is the displacement.

2. **Analyze the Given Function**:
- Let's consider the function given in the question:
\[ y = \sin(\omega t) - \cos(\omega t) \]

3. **Calculate the Velocity**:
- The velocity \( v \) is the first derivative of displacement \( y \) with respect to time \( t \):
\[ v = \frac{dy}{dt} = \frac{d}{dt}(\sin(\omega t) - \cos(\omega t)) \]
- Using the chain rule:
\[ v = \omega \cos(\omega t) + \omega \sin(\omega t) \]

4. **Calculate the Acceleration**:
- The acceleration \( a \) is the derivative of velocity \( v \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(\omega \cos(\omega t) + \omega \sin(\omega t)) \]
- Again, applying the chain rule:
\[ a = -\omega^2 \sin(\omega t) + \omega^2 \cos(\omega t) \]
- This can be rewritten as:
\[ a = \omega^2(-\sin(\omega t) + \cos(\omega t)) \]

5. **Check the Proportionality**:
- We can factor out \( -\omega^2 \):
\[ a = -\omega^2(\sin(\omega t) - \cos(\omega t)) \]
- Notice that \( y = \sin(\omega t) - \cos(\omega t) \), which means:
\[ a = -\omega^2 y \]
- This satisfies the condition for SHM since \( a \) is proportional to \( -y \).

6. **Conclusion**:
- Since the derived acceleration is proportional to the negative of the displacement, the function \( y = \sin(\omega t) - \cos(\omega t) \) represents simple harmonic motion.

### Final Answer:
The function that represents simple harmonic oscillation is:
\[ y = \sin(\omega t) - \cos(\omega t) \]