Two identical springs are fixed at one end and masses `1kg` and `4kg` are suspended at their other ends. They are both strethced down form their mean position and let go simultaneously. If they are in the same phase after every `4` seconds then the springs constant `k` is

Options

A. `pi(N)/(m)`
B. `pi^(2)(N)/(m)`
C. `2pi(N)/(m)`
D. given data is insufficient

Explanation

To solve the problem, we need to determine the spring constant \( k \) based on the information provided about the two masses and their oscillation phases.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two identical springs with masses \( m_1 = 1 \, \text{kg} \) and \( m_2 = 4 \, \text{kg} \) attached to them.
- Both masses are released from their mean position simultaneously.

2. **Angular Frequency Calculation**:
- The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
- For the mass \( m_1 = 1 \, \text{kg} \):
\[ \omega_1 = \sqrt{\frac{k}{1}} = \sqrt{k} \]
- For the mass \( m_2 = 4 \, \text{kg} \):
\[ \omega_2 = \sqrt{\frac{k}{4}} = \frac{\sqrt{k}}{2} \]

3. **Phase of the Masses**:
- The phase of the first mass can be expressed as:
\[ \text{Phase}_1 = \omega_1 t = \sqrt{k} t \]
- The phase of the second mass can be expressed as:
\[ \text{Phase}_2 = \omega_2 t = \frac{\sqrt{k}}{2} t \]

4. **Condition for Same Phase**:
- The problem states that the two masses are in the same phase after every 4 seconds. This means the phase difference must be an integer multiple of \( 2\pi \):
\[ \text{Phase}_1 - \text{Phase}_2 = n \cdot 2\pi \]
- Substituting the expressions for the phases:
\[ \sqrt{k} t - \frac{\sqrt{k}}{2} t = n \cdot 2\pi \]
- Simplifying this gives:
\[ \left(\sqrt{k} - \frac{\sqrt{k}}{2}\right) t = n \cdot 2\pi \]
\[ \frac{\sqrt{k}}{2} t = n \cdot 2\pi \]

5. **Finding the Time Interval**:
- We know that the time interval \( t \) for the same phase is given as 4 seconds:
\[ \frac{\sqrt{k}}{2} \cdot 4 = n \cdot 2\pi \]
- Rearranging gives:
\[ \sqrt{k} = \frac{n \cdot 2\pi}{2} \]
\[ \sqrt{k} = n \cdot \pi \]

6. **Determining \( k \)**:
- The first time they are in phase (for \( n = 1 \)):
\[ \sqrt{k} = \pi \implies k = \pi^2 \]

### Conclusion:
The spring constant \( k \) is \( \pi^2 \).