A spring of force constant k is cut into two pieces whose lengths are in the ratio 1:2. The force constant of the longer piece?

Options

A. `k//2`
B. `3k//2`
C. `2k`
D. `3k`

Explanation

To find the force constant of the longer piece of the spring when it is cut into two pieces in the ratio of 1:2, we can follow these steps:

### Step-by-Step Solution:

1. **Define the lengths of the pieces**:
Let the length of the first piece be \( L_1 = x \) and the length of the second piece be \( L_2 = 2x \). The total length of the spring is then:
\[ L = L_1 + L_2 = x + 2x = 3x \]

2. **Understand the relationship between force constant and length**:
The force constant \( k \) of a spring is inversely proportional to its length. This means:
\[ k \propto \frac{1}{L} \]
Therefore, if we denote the force constant of the original spring as \( k \), we can express the force constant of a piece of spring as:
\[ k' = \frac{c}{L} \]
where \( c \) is a constant.

3. **Relate the constant \( c \) to the original spring**:
For the original spring of length \( L \) and force constant \( k \):
\[ c = k \cdot L \]

4. **Calculate the force constant of the longer piece**:
The length of the longer piece \( L_2 = 2x \). We can substitute \( c \) and \( L_2 \) into the formula for the force constant:
\[ k' = \frac{c}{L_2} = \frac{k \cdot L}{2x} \]

5. **Substitute \( x \) in terms of \( L \)**:
From the earlier step, we found that \( L = 3x \), which gives us \( x = \frac{L}{3} \). Substituting this into the equation for \( k' \):
\[ k' = \frac{k \cdot L}{2 \cdot \frac{L}{3}} = \frac{k \cdot L}{\frac{2L}{3}} = \frac{3k}{2} \]

6. **Conclusion**:
The force constant of the longer piece \( k' \) is:
\[ k' = \frac{3k}{2} \]

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

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