The PE of a certain spring when streched from natural length through a distance `0.3m` is `5.6J`. Find the amount of work in joule that must be done on this spring to stretch it through an additional distance `0.15m`.

Explanation

To solve the problem, we need to find the work done on a spring when it is stretched an additional distance of 0.15 m, given that the potential energy (PE) when stretched to 0.3 m is 5.6 J.

### Step-by-Step Solution:

1. **Understand the Potential Energy of the Spring**:
The potential energy (PE) stored in a spring when stretched by a distance \( x \) is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the distance stretched from the natural length.

2. **Calculate the Spring Constant \( k \)**:
We know that when \( x = 0.3 \, m \), the potential energy \( U = 5.6 \, J \). Plugging these values into the formula:
\[ 5.6 = \frac{1}{2} k (0.3)^2 \]
Rearranging this equation to solve for \( k \):
\[ k = \frac{2 \times 5.6}{(0.3)^2} \]
\[ k = \frac{11.2}{0.09} = 124.44 \, N/m \]

3. **Determine the New Stretch Distance**:
The new stretch distance will be:
\[ x_{final} = 0.3 \, m + 0.15 \, m = 0.45 \, m \]

4. **Calculate the Potential Energy at the New Stretch Distance**:
Now, we calculate the potential energy when the spring is stretched to \( 0.45 \, m \):
\[ PE_{final} = \frac{1}{2} k (0.45)^2 \]
Substituting the value of \( k \):
\[ PE_{final} = \frac{1}{2} \times 124.44 \times (0.45)^2 \]
\[ PE_{final} = \frac{1}{2} \times 124.44 \times 0.2025 = 12.59 \, J \]

5. **Calculate the Work Done to Stretch the Spring**:
The work done on the spring to stretch it from \( 0.3 \, m \) to \( 0.45 \, m \) is the change in potential energy:
\[ Work = PE_{final} - PE_{initial} \]
\[ Work = 12.59 - 5.6 = 6.99 \, J \]

### Final Answer:
The amount of work that must be done on the spring to stretch it through an additional distance of \( 0.15 \, m \) is approximately \( 6.99 \, J \).