A spring fo spring constant k is broken in the length ratio 1:3. The spring constant of larger part will be

Options

A. 4k/3
B. 2k/3
C. k/3
D. 5k/3

Explanation

To find the spring constant of the larger part of a broken spring, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a spring with a spring constant \( k \) that is broken into two parts in the ratio of 1:3. This means if the total length of the spring is \( L \), the lengths of the two parts will be \( \frac{L}{4} \) and \( \frac{3L}{4} \).

2. **Identify the Lengths**:
- Let the total length of the spring be \( L \).
- The length of the first part (smaller part) is \( \frac{L}{4} \).
- The length of the second part (larger part) is \( \frac{3L}{4} \).

3. **Use the Relationship Between Spring Constant and Length**:
The spring constant \( k \) is inversely proportional to the length of the spring when the material and the number of coils remain the same. The relationship can be expressed as:
\[ k \cdot L = k_1 \cdot L_1 = k_2 \cdot L_2 \]
where \( k_1 \) and \( k_2 \) are the spring constants of the smaller and larger parts, respectively, and \( L_1 \) and \( L_2 \) are their lengths.

4. **Set Up the Equations**:
- For the smaller part:
\[ k \cdot L = k_1 \cdot \frac{L}{4} \]
- For the larger part:
\[ k \cdot L = k_2 \cdot \frac{3L}{4} \]

5. **Solve for \( k_1 \)**:
From the first equation:
\[ k_1 = \frac{k \cdot L}{\frac{L}{4}} = 4k \]

6. **Solve for \( k_2 \)**:
From the second equation:
\[ k_2 = \frac{k \cdot L}{\frac{3L}{4}} = \frac{4k}{3} \]

7. **Conclusion**: The spring constant of the larger part of the spring is:
\[ k_2 = \frac{4k}{3} \]

### Final Answer:
The spring constant of the larger part will be \( \frac{4k}{3} \).

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