A spring of certain length and having spring constant `k` is cut into two pieces of length in a ratio `1:2`. The spring constants of the two pieces are in a ratio `:`

Options

A. `1:1`
B. `1:4`
C. `1:2`
D. `2:1`

Explanation

To solve the problem, we need to analyze how the spring constant changes when a spring is cut into two pieces.

### Step-by-Step Solution:

1. **Understanding the Spring Constant**:
The spring constant \( k \) is defined as the force required to stretch or compress the spring by a unit length. For a spring of length \( L \), the spring constant \( k \) is inversely proportional to its length. Mathematically, this can be expressed as:
\[ k \propto \frac{1}{L} \]

2. **Dividing the Spring**:
We have a spring of total length \( L \) that is cut into two pieces in the ratio \( 1:2 \). Let the lengths of the two pieces be \( L_1 \) and \( L_2 \). Since the ratio is \( 1:2 \), we can express the lengths as:
\[ L_1 = \frac{L}{3} \quad \text{and} \quad L_2 = \frac{2L}{3} \]

3. **Finding the Spring Constants**:
Using the relationship \( k \propto \frac{1}{L} \), we can find the spring constants for the two pieces:
- For the first piece of length \( L_1 \):
\[ k_1 \propto \frac{1}{L_1} = \frac{1}{\frac{L}{3}} = \frac{3}{L} \]
- For the second piece of length \( L_2 \):
\[ k_2 \propto \frac{1}{L_2} = \frac{1}{\frac{2L}{3}} = \frac{3}{2L} \]

4. **Calculating the Ratio of Spring Constants**:
Now, we can find the ratio of the spring constants \( k_1 \) and \( k_2 \):
\[ \frac{k_1}{k_2} = \frac{\frac{3}{L}}{\frac{3}{2L}} = \frac{3}{L} \times \frac{2L}{3} = 2 \]
Therefore, the ratio of the spring constants is:
\[ k_1 : k_2 = 2 : 1 \]

### Final Answer:
The spring constants of the two pieces are in the ratio \( 2:1 \).