A certain spring that obeys Hook's law is stretched by an external agents. The work done in stretching the spring by 10 cm is 4 J . How much additional work is required to stretch the spring an additional `10 cm`?

Options

A. `2 J`
B. `4 J`
C. `8 J`
D. `12 J`

Explanation

To solve the problem of how much additional work is required to stretch the spring an additional 10 cm after it has already been stretched by 10 cm, we can follow these steps:

### Step 1: Understand the Work Done on the Spring
According to Hooke's Law, the work done (W) in stretching a spring is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the amount of stretch.

### Step 2: Calculate the Spring Constant (k)
From the problem, we know that the work done to stretch the spring by 10 cm (0.1 m) is 4 J. We can use this information to find the spring constant \( k \).

Substituting the known values into the work formula:
\[ 4 = \frac{1}{2} k (0.1)^2 \]

This simplifies to:
\[ 4 = \frac{1}{2} k (0.01) \]
\[ 4 = 0.005 k \]

Now, solving for \( k \):
\[ k = \frac{4}{0.005} = 800 \, \text{N/m} \]

### Step 3: Calculate the Work Done to Stretch the Spring to 20 cm
Now we need to calculate the work done to stretch the spring to 20 cm (0.2 m). Using the same formula:
\[ W' = \frac{1}{2} k (0.2)^2 \]

Substituting the value of \( k \):
\[ W' = \frac{1}{2} (800) (0.2)^2 \]
\[ W' = \frac{1}{2} (800) (0.04) \]
\[ W' = 400 \times 0.04 = 16 \, \text{J} \]

### Step 4: Calculate the Additional Work Required
The additional work required to stretch the spring from 10 cm to 20 cm is the difference between the work done to stretch it to 20 cm and the work done to stretch it to 10 cm:
\[ \text{Additional Work} = W' - W \]
\[ \text{Additional Work} = 16 \, \text{J} - 4 \, \text{J} = 12 \, \text{J} \]

### Final Answer
The additional work required to stretch the spring an additional 10 cm is **12 J**.

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