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 of finding the force constant of the longer piece of the spring after it has been cut, we can follow these steps:

### Step 1: Define the lengths of the pieces
Let the total length of the original spring be \( L \). When the spring is cut into two pieces such that one piece is double the length of the other, we can denote the lengths of the two pieces as follows:
- Let the length of the shorter piece be \( x \).
- Therefore, the length of the longer piece will be \( 2x \).

### Step 2: Relate the lengths to the total length
Since the total length of the spring is \( L \), we can write the equation:
\[ x + 2x = L \]
This simplifies to:
\[ 3x = L \]
From this, we can find \( x \):
\[ x = \frac{L}{3} \]
Thus, the lengths of the two pieces are:
- Shorter piece: \( x = \frac{L}{3} \)
- Longer piece: \( 2x = \frac{2L}{3} \)

### Step 3: Understand the relationship between force constant and length
The force constant \( k \) of a spring is inversely proportional to its length. The relationship can be expressed as:
\[ k' = \frac{k}{\text{length of the spring}} \]
where \( k' \) is the new force constant after cutting the spring.

### Step 4: Calculate the new force constant for the longer piece
For the longer piece, which has a length of \( \frac{2L}{3} \), we can find its new force constant \( k_{long} \):
\[ k_{long} = \frac{k}{\frac{2L}{3}} = k \cdot \frac{3}{2L} \]

### Step 5: Relate the original force constant to the new one
Since the original spring had a force constant \( k \) when it had a length \( L \), we can express the original force constant as:
\[ k = \frac{k}{L} \]
Thus, substituting back into our equation for \( k_{long} \):
\[ k_{long} = k \cdot \frac{3}{2} \]

### Conclusion
The force constant of the longer piece of the spring is:
\[ k_{long} = \frac{3k}{2} \]