A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will

Options

A. remain T
B. become 2T
C. become T/2
D. become `T/sqrt2`

Answer

To solve the problem, we need to analyze how the time period of a spring-mass system changes when the spring is cut in half. Let's go through the steps systematically.

### Step-by-Step Solution

1. **Understand the Initial Time Period**:
The time period \( T \) of a mass-spring system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass attached to the spring and \( k \) is the spring constant.

2. **Cutting the Spring**:
When the spring of length \( L \) is cut into two equal parts, each part has a length of \( \frac{L}{2} \).

3. **Finding the New Spring Constant**:
The spring constant \( k' \) of the new spring can be determined using the relationship between the length of the spring and its spring constant. The product of the length and spring constant remains constant:
\[ L \cdot k = \frac{L}{2} \cdot k' \]
Rearranging gives:
\[ k' = 2k \]

4. **Calculating the New Time Period**:
Now, we substitute \( k' \) into the formula for the time period of the new spring:
\[ T' = 2\pi \sqrt{\frac{m}{k'}} \]
Substituting \( k' = 2k \):
\[ T' = 2\pi \sqrt{\frac{m}{2k}} = 2\pi \sqrt{\frac{m}{k}} \cdot \frac{1}{\sqrt{2}} = \frac{T}{\sqrt{2}} \]

5. **Final Result**:
Therefore, the new time period \( T' \) when one half of the spring is used for simple harmonic motion is:
\[ T' = \frac{T}{\sqrt{2}} \]

### Conclusion
The correct answer is that the new time period becomes \( \frac{T}{\sqrt{2}} \).