A particle moves such that its accleration a is given by `a = -bx`, where x is the displacement from equilibrium position and b is a constant. The period of oscillation is

Options

A. `sqrt(2pi) //b`
B. `2pi//sqrt(b)`
C. `2pi//b`
D. `2sqrt((pi//b))`

Explanation

To solve the problem, we need to determine the period of oscillation for a particle whose acceleration is given by the equation \( a = -bx \), where \( x \) is the displacement from the equilibrium position and \( b \) is a constant.

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
The acceleration \( a \) is given by \( a = -bx \). This indicates that the acceleration is directly proportional to the displacement \( x \) but acts in the opposite direction (hence the negative sign). This is characteristic of simple harmonic motion (SHM).

2. **Identifying the Nature of Motion**:
Since the acceleration is proportional to the displacement and directed towards the equilibrium position, we can conclude that the particle is undergoing simple harmonic motion (SHM).

3. **Using the Formula for Period in SHM**:
The formula for the period \( T \) of a simple harmonic oscillator is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass of the particle and \( k \) is the spring constant. In our case, we can relate the given acceleration to the standard form of SHM.

4. **Relating Acceleration to SHM**:
In SHM, we have:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency. Comparing this with our equation \( a = -bx \), we can identify:
\[ \omega^2 = b \]

5. **Finding the Period**:
The angular frequency \( \omega \) is related to the period \( T \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting \( \omega = \sqrt{b} \) into the period formula gives us:
\[ T = \frac{2\pi}{\sqrt{b}} \]

6. **Final Expression for the Period**:
Therefore, the period of oscillation \( T \) is:
\[ T = 2\pi \sqrt{\frac{1}{b}} \]

### Conclusion:
The period of oscillation for the particle is given by:
\[ T = 2\pi \sqrt{\frac{1}{b}} \]