A spring 40 mm long is stretched by the application of a force. If 10 N force required to stretch the spring through 1mm, then the work done in stretching the spring through 40 mm is

Options

A. 84 J
B. 68 J
C. 23 J
D. 8 J

Explanation

To solve the problem of calculating the work done in stretching a spring, we can follow these steps:

### Step 1: Identify the given values
- The spring is initially 40 mm long.
- A force of 10 N is required to stretch the spring by 1 mm.

### Step 2: Convert units
- Convert the stretch from mm to meters for consistency in SI units.
- 1 mm = 0.001 m
- Therefore, 40 mm = 40 * 0.001 = 0.04 m.

### Step 3: Calculate the spring constant (k)
- According to Hooke's Law, the force (F) applied to stretch the spring is proportional to the displacement (x):
\[ F = kx \]
- Here, \( F = 10 \, \text{N} \) and \( x = 1 \, \text{mm} = 0.001 \, \text{m} \).
- Rearranging the formula to find k:
\[ k = \frac{F}{x} = \frac{10 \, \text{N}}{0.001 \, \text{m}} = 10,000 \, \text{N/m} = 10^4 \, \text{N/m} \]

### Step 4: Use the work done formula for springs
- The work done (W) in stretching a spring is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
- Substitute \( k = 10^4 \, \text{N/m} \) and \( x = 0.04 \, \text{m} \):
\[ W = \frac{1}{2} \times 10^4 \times (0.04)^2 \]

### Step 5: Calculate \( (0.04)^2 \)
- Calculate \( (0.04)^2 \):
\[ (0.04)^2 = 0.0016 \]

### Step 6: Substitute and calculate the work done
- Now substitute back into the work done formula:
\[ W = \frac{1}{2} \times 10^4 \times 0.0016 \]
\[ W = \frac{1}{2} \times 10^4 \times 1.6 \times 10^{-3} \]
\[ W = \frac{1}{2} \times 16 = 8 \, \text{Joules} \]

### Final Answer
The work done in stretching the spring through 40 mm is **8 Joules**.

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