The potential energy of a certain spring when stretched through a distance 'S' is `10` joule. The amount of work (in joule) that must be done on this spring to stretch it through an additional distance 'S' will be

Options

A. 10 J
B. 20 J
C. 30 J
D. 40 J

Explanation

To solve the problem, we need to calculate the work done on the spring when it is stretched through an additional distance 'S'. We will use the formula for the potential energy stored in a spring and the concept of work done.

### Step-by-Step Solution:

1. **Understand the Potential Energy Formula**:
The potential energy (U) stored in a spring when it is stretched by a distance 'x' is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant.

2. **Calculate Potential Energy for Distance 'S'**:
According to the problem, when the spring is stretched through a distance 'S', the potential energy is given as:
\[ U = 10 \text{ joules} \]
Thus, we can write:
\[ \frac{1}{2} k S^2 = 10 \]

3. **Calculate Potential Energy for Distance '2S'**:
Now, we need to find the potential energy when the spring is stretched through an additional distance 'S', making the total distance '2S':
\[ U' = \frac{1}{2} k (2S)^2 \]
Simplifying this, we get:
\[ U' = \frac{1}{2} k (4S^2) = 2 k S^2 \]

4. **Relate \( U' \) to \( U \)**:
We know from step 2 that:
\[ \frac{1}{2} k S^2 = 10 \text{ joules} \]
Therefore:
\[ k S^2 = 20 \text{ joules} \]
Substituting this into the equation for \( U' \):
\[ U' = 2 \times 20 = 40 \text{ joules} \]

5. **Calculate the Work Done**:
The work done (W) on the spring to stretch it from 'S' to '2S' is the change in potential energy:
\[ W = U' - U \]
Substituting the values we found:
\[ W = 40 \text{ joules} - 10 \text{ joules} = 30 \text{ joules} \]

### Final Answer:
The amount of work that must be done on this spring to stretch it through an additional distance 'S' is **30 joules**.