A block with a mass of `3.00 kg` is suspended from an ideal spring having negligible mass and stretches the spring by `0.2 m`.
(a) What is the force constant of the spring?
(b) What is the period of oscillation of the block if it is pulled down and released ?

Explanation

To solve the given problem, we will break it down into two parts: (a) finding the force constant of the spring and (b) calculating the period of oscillation of the block.

### Part (a): Finding the Force Constant of the Spring

1. **Identify the forces acting on the block**:
- The weight of the block (downward force) is given by \( F_g = mg \).
- The spring force (upward force) is given by \( F_s = kx \), where \( k \) is the spring constant and \( x \) is the extension of the spring.

2. **Set up the equilibrium condition**:
At equilibrium, the downward force (weight of the block) is equal to the upward force (spring force):
\[ mg = kx \]

3. **Rearrange the equation to find the spring constant \( k \)**:
\[ k = \frac{mg}{x} \]

4. **Substitute the known values**:
- Mass \( m = 3.00 \, \text{kg} \)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)
- Extension \( x = 0.2 \, \text{m} \)

Plugging in these values:
\[ k = \frac{3.00 \, \text{kg} \times 9.8 \, \text{m/s}^2}{0.2 \, \text{m}} = \frac{29.4 \, \text{N}}{0.2 \, \text{m}} = 147 \, \text{N/m} \]

5. **Conclusion for part (a)**:
The force constant of the spring is \( k = 147 \, \text{N/m} \).

### Part (b): Finding the Period of Oscillation

1. **Use the formula for the period of oscillation**:
The period \( T \) of a mass-spring system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

2. **Substitute the known values**:
- Mass \( m = 3.00 \, \text{kg} \)
- Spring constant \( k = 147 \, \text{N/m} \)

Plugging in these values:
\[ T = 2\pi \sqrt{\frac{3.00 \, \text{kg}}{147 \, \text{N/m}}} \]

3. **Calculate the value inside the square root**:
\[ \frac{3.00}{147} \approx 0.02041 \]

4. **Calculate the square root**:
\[ \sqrt{0.02041} \approx 0.1429 \]

5. **Calculate the period \( T \)**:
\[ T = 2\pi \times 0.1429 \approx 0.897 \, \text{s} \]

6. **Conclusion for part (b)**:
The period of oscillation of the block is approximately \( T \approx 0.897 \, \text{s} \).

### Summary of Answers:
(a) The force constant of the spring is \( k = 147 \, \text{N/m} \).
(b) The period of oscillation of the block is \( T \approx 0.897 \, \text{s} \).