Time period of a spring mass system is T.If this spring is cut into two parts whose lengths are in ratio 1:3 and the same mass is attached to the longer part, the new time period will be

Options

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

Explanation

To solve the problem, we need to find the new time period of a spring-mass system after the spring is cut into two parts in the ratio 1:3, and the same mass is attached to the longer part. Here’s a step-by-step solution:

### Step 1: Understand the Time Period Formula
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.

### Step 2: Determine the Lengths of the Spring Parts
Let the total length of the spring be \( L \). According to the problem, the spring is divided into two parts in the ratio 1:3. Therefore:
- Length of the first part \( L_1 = \frac{L}{4} \)
- Length of the second part \( L_2 = \frac{3L}{4} \)

### Step 3: Find the Spring Constant of the Longer Part
The spring constant \( k \) is inversely proportional to the length of the spring. Thus, if the original spring constant is \( k \) for length \( L \):
\[ k_1 = \frac{kL}{L_1} = \frac{kL}{\frac{L}{4}} = 4k \]
for the shorter part, and for the longer part \( L_2 \):
\[ k_2 = \frac{kL}{L_2} = \frac{kL}{\frac{3L}{4}} = \frac{4k}{3} \]

### Step 4: Calculate the New Time Period
Now, we will calculate the new time period \( T' \) using the spring constant \( k_2 \):
\[ T' = 2\pi \sqrt{\frac{m}{k_2}} = 2\pi \sqrt{\frac{m}{\frac{4k}{3}}} = 2\pi \sqrt{\frac{3m}{4k}} \]

### Step 5: Relate the New Time Period to the Original Time Period
We know from the original time period \( T \):
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Thus, we can express \( T' \) in terms of \( T \):
\[ T' = T \cdot \sqrt{\frac{3}{4}} = T \cdot \frac{\sqrt{3}}{2} \]

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

### Summary
The new time period of the spring-mass system after cutting the spring into two parts in the ratio 1:3 and attaching the mass to the longer part is \( T' = T \cdot \frac{\sqrt{3}}{2} \).

---