A spring when compressed by `4 cm` has `2 J` energy stored in it. The force requried to extend it by `8 cm` will be

Options

A. 20 N
B. 2 N
C. 200 N
D. 2000 N

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between energy stored in a spring and its compression/extension.
The energy (U) stored in a spring is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where:
- \( U \) = energy stored in the spring (in Joules)
- \( k \) = spring constant (in N/m)
- \( x \) = compression or extension of the spring from its equilibrium position (in meters)

### Step 2: Use the given information to find the spring constant \( k \).
We know that when the spring is compressed by \( 4 \, \text{cm} \) (which is \( 0.04 \, \text{m} \)), the energy stored is \( 2 \, \text{J} \). We can plug these values into the energy formula to find \( k \):
\[ 2 = \frac{1}{2} k (0.04)^2 \]

### Step 3: Solve for \( k \).
Rearranging the equation, we get:
\[ 2 = \frac{1}{2} k (0.0016) \]
\[ 2 = 0.0008 k \]
Now, solving for \( k \):
\[ k = \frac{2}{0.0008} \]
\[ k = 2500 \, \text{N/m} \]

### Step 4: Calculate the force required to extend the spring by \( 8 \, \text{cm} \).
The force \( F \) required to extend the spring by \( x \) meters is given by Hooke's Law:
\[ F = kx \]
Here, \( x = 8 \, \text{cm} = 0.08 \, \text{m} \). Now substituting the values:
\[ F = 2500 \times 0.08 \]

### Step 5: Solve for \( F \).
Calculating the force:
\[ F = 2500 \times 0.08 = 200 \, \text{N} \]

### Final Answer:
The force required to extend the spring by \( 8 \, \text{cm} \) is \( 200 \, \text{N} \).

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