The equation of motion of a simple harmonic motion is

Options

A. `(d^(2)x)/(dt^(2))=-omega^(2)x`
B. `(d^(2)x)/(dt^(2))=-omega^(2)t`
C. `(d^(2)x)/(dt^(2))=-omegax`
D. `(d^(2)x)/(dt^(2))=-omegat`

Explanation

To solve the problem of finding the equation of motion for simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of SHM**:
Simple Harmonic Motion is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

2. **Write the General Equation of SHM**:
The displacement \( x \) in SHM can be expressed as:
\[ x(t) = A \cos(\omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

3. **Differentiate to Find Velocity**:
To find the velocity \( v \), differentiate \( x(t) \) with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \]

4. **Differentiate Again to Find Acceleration**:
Now, differentiate the velocity to find acceleration \( a \):
\[ a(t) = \frac{d^2x}{dt^2} = -A \omega^2 \cos(\omega t + \phi) \]

5. **Relate Acceleration to Displacement**:
Since \( x(t) = A \cos(\omega t + \phi) \), we can substitute this back into the equation for acceleration:
\[ a(t) = -\omega^2 x(t) \]
This can be rewritten as:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

6. **Conclusion**:
The equation of motion for simple harmonic motion is:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]
This indicates that the acceleration is proportional to the displacement and directed towards the equilibrium position.

### Final Answer:
The equation of motion of simple harmonic motion is:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]