If the force constant of a wire is K, the work done in increasing the length of the wire by `l` is

Options

A. `K//2`
B. `Kl`
C. `Kl^(2)//2`
D. `Kl^(2)`

Answer

To find the work done in increasing the length of a wire by a distance \( l \) when the force constant (spring constant) of the wire is \( K \), we can follow these steps:

### Step 1: Understand the relationship between force and extension
The force constant \( K \) of a wire is defined as:
\[ K = \frac{F}{l} \]
where \( F \) is the force applied to stretch the wire and \( l \) is the extension (increase in length) of the wire.

### Step 2: Express the force in terms of the force constant and extension
From the equation above, we can rearrange it to express the force \( F \):
\[ F = K \cdot l \]

### Step 3: Use the formula for work done
The work done \( W \) in stretching the wire can be expressed as the potential energy stored in the wire when it is stretched. The formula for the potential energy (or work done) in stretching a spring (or wire) is given by:
\[ W = \frac{1}{2} \cdot F \cdot l \]

### Step 4: Substitute the expression for force into the work done formula
Now substitute \( F = K \cdot l \) into the work done formula:
\[ W = \frac{1}{2} \cdot (K \cdot l) \cdot l \]

### Step 5: Simplify the expression
This simplifies to:
\[ W = \frac{1}{2} \cdot K \cdot l^2 \]

### Final Answer
Thus, the work done in increasing the length of the wire by \( l \) is:
\[ W = \frac{1}{2} K l^2 \]