The time period of a spring - mass system is T. If this spring is cut into two parts, whose lengths are in the ratio `1:2`, and the same mass is attached to the longer part, the new time period will be

Options

A. `sqrt((1)/(3))T`
B. `sqrt((2)/(3))T`
C. `(sqrt3T)/(2)`
D. `sqrt3T`

Answer

To solve the problem, we need to determine the new time period of a spring-mass system after the spring is cut into two parts in the ratio of 1:2, and the same mass is attached to the longer part.

### Step-by-Step Solution:

1. **Understand the Time Period of a Spring-Mass System**:
The time period \( T \) of a spring-mass 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**:
We are given that the spring is cut into two parts in the ratio \( 1:2 \). Let the total length of the spring be \( L \). Then, the lengths of the two parts can be defined as:
- \( L_1 = \frac{L}{3} \) (shorter part)
- \( L_2 = \frac{2L}{3} \) (longer part)

3. **Finding the Spring Constant**:
The spring constant \( k \) is inversely proportional to the length of the spring. If the original spring constant is \( k \), then for the shorter part \( k_1 \) and the longer part \( k_2 \), we have:
\[ k_1 L_1 = k_2 L_2 \]
Rearranging gives:
\[ k_2 = \frac{k_1 L_1}{L_2} \]
Since \( L_1 = \frac{L}{3} \) and \( L_2 = \frac{2L}{3} \), substituting these into the equation gives:
\[ k_2 = \frac{k \cdot \frac{L}{3}}{\frac{2L}{3}} = \frac{k}{2} \]

4. **Calculating the New Time Period**:
Now, we attach the same mass \( m \) to the longer part of the spring, which has the spring constant \( k_2 = \frac{3}{2}k \). The new time period \( T' \) is given by:
\[ T' = 2\pi \sqrt{\frac{m}{k_2}} = 2\pi \sqrt{\frac{m}{\frac{3}{2}k}} = 2\pi \sqrt{\frac{2m}{3k}} \]

5. **Relating New Time Period to Original Time Period**:
We can relate this new time period to the original time period \( T \):
\[ T' = \sqrt{\frac{2}{3}} T \]

### Final Answer:
The new time period \( T' \) of the spring-mass system when the mass is attached to the longer part of the spring is:
\[ T' = \sqrt{\frac{2}{3}} T \]