Three mass 0.1 kg ,0.3 kg and 0.4 kg are suspended at end of a spring. When is 0.4 kg mass is removed , the system oscillates with a period 2 s . When the 0.3 kg mass is also removed , the system will oscillates with a period

Options

A. 1 s
B. 2 s
C. 3 s
D. 4 s

Explanation

To solve the problem step by step, we will determine the period of oscillation when the 0.3 kg mass is also removed from the spring-mass system.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- We have three masses: \( m_1 = 0.1 \, \text{kg} \), \( m_2 = 0.3 \, \text{kg} \), and \( m_3 = 0.4 \, \text{kg} \).
- When the 0.4 kg mass is removed, the remaining mass is \( m_1 + m_2 = 0.1 \, \text{kg} + 0.3 \, \text{kg} = 0.4 \, \text{kg} \).

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

3. **Set up the equation for the first scenario**:
- When the 0.4 kg mass is removed, the period is given as \( T = 2 \, \text{s} \).
- Thus, we have:
\[ 2 = 2\pi \sqrt{\frac{0.4}{k}} \]
- Dividing both sides by \( 2\pi \):
\[ \frac{1}{\pi} = \sqrt{\frac{0.4}{k}} \]
- Squaring both sides:
\[ \frac{1}{\pi^2} = \frac{0.4}{k} \]
- Rearranging gives:
\[ k = 0.4\pi^2 \]

4. **Set up the equation for the second scenario**:
- Now, we need to find the period when the 0.3 kg mass is also removed, leaving only the 0.1 kg mass.
- The new mass \( m = 0.1 \, \text{kg} \).

5. **Calculate the new period**:
- Using the same formula for the period:
\[ T = 2\pi \sqrt{\frac{0.1}{k}} \]
- Substituting \( k = 0.4\pi^2 \):
\[ T = 2\pi \sqrt{\frac{0.1}{0.4\pi^2}} \]
- Simplifying:
\[ T = 2\pi \sqrt{\frac{0.1}{0.4}} \cdot \frac{1}{\pi} = 2\sqrt{\frac{0.1}{0.4}} = 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1 \, \text{s} \]

6. **Conclusion**:
- The period of oscillation when the 0.3 kg mass is also removed is \( T = 1 \, \text{s} \).

### Final Answer:
The period of oscillation when the 0.3 kg mass is also removed is **1 second**.