A spring of force constant `800N//m` has an extension of 5cm. The work done in extending it from 5cm to 15cm is

Options

A. (a) 16J
B. (b) 8J
C. (c) 32J
D. (d) 24J

Explanation

To find the work done in extending a spring from 5 cm to 15 cm, we can use Hooke's law and the formula for work done on a spring. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have a spring with a force constant (k) of 800 N/m, and we want to calculate the work done in extending it from an initial extension of 5 cm to a final extension of 15 cm.

### Step 2: Convert units
Since the spring constant is given in N/m, we need to convert the extensions from centimeters to meters:
- Initial extension (x1) = 5 cm = 0.05 m
- Final extension (x2) = 15 cm = 0.15 m

### Step 3: Use the work done formula
The work done (W) in stretching or compressing a spring is given by the formula:
\[ W = \frac{1}{2} k (x_2^2 - x_1^2) \]
Where:
- \( k \) is the spring constant,
- \( x_1 \) is the initial extension,
- \( x_2 \) is the final extension.

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ W = \frac{1}{2} \times 800 \, \text{N/m} \times ((0.15)^2 - (0.05)^2) \]

### Step 5: Calculate the squares
Calculate \( (0.15)^2 \) and \( (0.05)^2 \):
\[ (0.15)^2 = 0.0225 \]
\[ (0.05)^2 = 0.0025 \]

### Step 6: Find the difference
Now, find the difference:
\[ 0.0225 - 0.0025 = 0.02 \]

### Step 7: Substitute back into the work formula
Now substitute this difference back into the work formula:
\[ W = \frac{1}{2} \times 800 \times 0.02 \]

### Step 8: Calculate the work done
\[ W = 400 \times 0.02 = 8 \, \text{J} \]

### Final Answer
The work done in extending the spring from 5 cm to 15 cm is **8 Joules**.

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