The equation of motion of a particle executing SHM is (k is a positive constant)

Options

A. `(d^2 x)/(dt^2)- (k)/(m) x=0`
B. `(dx)/(dt)- (k)/(m) x=0`
C. `((dx)/(dt))^2 - (k)/(m) x=0`
D. `(d^2 x)/(dt^2)+ (k)/(m) x=0`

Answer

To solve the problem of finding the equation of motion for a particle executing Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding SHM**:
- In SHM, the acceleration \( a \) of a particle is directly proportional to its displacement \( x \) from the mean position and is directed towards the mean position. This can be expressed mathematically as:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency.

2. **Relating Acceleration to Displacement**:
- The acceleration \( a \) can also be expressed in terms of the second derivative of displacement with respect to time:
\[ a = \frac{d^2x}{dt^2} \]
- Therefore, we can equate the two expressions for acceleration:
\[ \frac{d^2x}{dt^2} = -\omega^2 x \]

3. **Substituting Angular Frequency**:
- The angular frequency \( \omega \) is related to the spring constant \( k \) and the mass \( m \) of the particle by the formula:
\[ \omega = \sqrt{\frac{k}{m}} \]
- Squaring this gives:
\[ \omega^2 = \frac{k}{m} \]

4. **Formulating the Equation of Motion**:
- Substituting \( \omega^2 \) into the equation of motion gives:
\[ \frac{d^2x}{dt^2} = -\frac{k}{m} x \]
- Rearranging this leads to the standard form of the equation of motion for SHM:
\[ \frac{d^2x}{dt^2} + \frac{k}{m} x = 0 \]

5. **Final Equation**:
- Therefore, the equation of motion of a particle executing SHM is:
\[ \frac{d^2x}{dt^2} + \frac{k}{m} x = 0 \]

6. **Identifying the Correct Option**:
- From the options provided, the correct equation that matches our derived equation is:
\[ \frac{d^2x}{dt^2} + \frac{k}{m} x = 0 \]
- Hence, the correct option is option D.