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

Options

A. (a) `(2/3)k`
B. (b) `(3/2)k`
C. (c) `3k`
D. (d) `6k`

Explanation

To solve the problem, we need to determine the spring constant of the longer piece after a spring of force constant \( k \) is cut into two pieces such that one piece is double the length of the other.

### Step-by-Step Solution:

1. **Identify the lengths of the pieces**:
Let the length of the shorter piece be \( L_1 \) and the longer piece be \( L_2 \). According to the problem, \( L_2 = 2L_1 \).

2. **Express the total length**:
The total length of the original spring is:
\[ L = L_1 + L_2 = L_1 + 2L_1 = 3L_1 \]
From this, we can express \( L_1 \) in terms of \( L \):
\[ L_1 = \frac{L}{3} \]

3. **Determine the length of the longer piece**:
Now substituting \( L_1 \) into the equation for \( L_2 \):
\[ L_2 = 2L_1 = 2 \left(\frac{L}{3}\right) = \frac{2L}{3} \]

4. **Relate the spring constants**:
The spring constant \( k \) for the original spring relates to the spring constants of the two pieces. The relationship between the spring constants and lengths is given by:
\[ k_1 L_1 = k_2 L_2 \]
where \( k_1 \) is the spring constant of the shorter piece and \( k_2 \) is the spring constant of the longer piece.

5. **Substituting the lengths**:
Substitute \( L_1 \) and \( L_2 \) into the equation:
\[ k_1 \left(\frac{L}{3}\right) = k_2 \left(\frac{2L}{3}\right) \]

6. **Simplifying the equation**:
Cancel \( \frac{L}{3} \) from both sides (assuming \( L \neq 0 \)):
\[ k_1 = 2k_2 \]

7. **Express \( k_1 \) in terms of \( k \)**:
Since the original spring has a spring constant \( k \), we can express \( k_1 \) in terms of \( k \). The spring constant of the shorter piece \( k_1 \) can be related to the original spring constant:
\[ k_1 = \frac{k \cdot L}{L_1} = \frac{k \cdot L}{\frac{L}{3}} = 3k \]

8. **Finding \( k_2 \)**:
Now substituting \( k_1 = 3k \) back into the equation \( k_1 = 2k_2 \):
\[ 3k = 2k_2 \implies k_2 = \frac{3k}{2} \]

9. **Conclusion**:
Therefore, the spring constant of the longer piece \( k_2 \) is:
\[ k_2 = \frac{3k}{2} \]

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