A spring force constant k is cut into two parts such that one piece is double the length of the other. Then the long piece will have a force constatnt of

Options

A. `(2//3)k`
B. `(3//2)k`
C. 3k
D. 6k

Explanation

To solve the problem of determining the spring constant of the longer piece when a spring is cut into two parts with one piece being double the length of the other, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Lengths of the Pieces**:
Let the total length of the spring be \( L \). We can denote the lengths of the two pieces as:
- Shorter piece: \( L_1 = \frac{L}{3} \)
- Longer piece: \( L_2 = \frac{2L}{3} \)

2. **Understand Spring Constants**:
The spring constant \( k \) of a spring is inversely proportional to its length when the material and thickness are constant. Therefore, we can express the spring constants of the two pieces as:
- Spring constant of the shorter piece: \( k_1 \)
- Spring constant of the longer piece: \( k_2 \)

The relationship between the spring constants and their lengths is given by:
\[ k_1 = \frac{k}{L_1} \quad \text{and} \quad k_2 = \frac{k}{L_2} \]

3. **Express the Spring Constants**:
Since \( L_1 = \frac{L}{3} \) and \( L_2 = \frac{2L}{3} \), we can substitute these into the equations:
\[ k_1 = \frac{k}{\frac{L}{3}} = \frac{3k}{L} \]
\[ k_2 = \frac{k}{\frac{2L}{3}} = \frac{3k}{2L} \]

4. **Combine the Springs in Series**:
When the two springs are connected in series, the equivalent spring constant \( K \) is given by:
\[ \frac{1}{K} = \frac{1}{k_1} + \frac{1}{k_2} \]

5. **Substitute the Spring Constants**:
Substitute \( k_1 \) and \( k_2 \) into the equation:
\[ \frac{1}{K} = \frac{L}{3k} + \frac{2L}{3k} \]
\[ \frac{1}{K} = \frac{L + 2L}{3k} = \frac{3L}{3k} = \frac{L}{k} \]

6. **Solve for the Equivalent Spring Constant**:
Therefore, we can find \( K \):
\[ K = \frac{k}{L} \]

7. **Determine the Spring Constant of the Longer Piece**:
Since we know that \( k_2 = \frac{3k}{2L} \), we can express it in terms of the original spring constant \( k \):
\[ k_2 = \frac{3k}{2} \]

Thus, the spring constant of the longer piece is \( \frac{3k}{2} \).

### Final Answer:
The spring constant of the longer piece is \( \frac{3k}{2} \).