To calculate the work done by the force, we need to use the concept of work from physics, which is defined as:

$W = \int F(x) \, dx$

Since the force varies with displacement, the work done is represented by the area under the force-displacement graph.

Step-by-Step Solution:

1. Look at the graph:

   • The force $F$ starts at $0 \, \text{N}$ when $x = 0 \, \text{m}$, increases linearly to $4 \, \text{N}$ at $x = 4 \, \text{m}$, and then decreases back to $0 \, \text{N}$ at $x = 10 \, \text{m}$.
   • This forms a triangle from $0 \, \text{m}$ to $10 \, \text{m}$, with a peak at $4 \, \text{m}$ and $4 \, \text{N}$.

2. Find the area of the triangle:

   • The area under the curve will give the total work done.
   • The base of the triangle is $10.0 \, \text{m}$ (displacement), and the height is $4.0 \, \text{N}$ (force).

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area} = \frac{1}{2} \times 10.0 \, \text{m} \times 4.0 \, \text{N} = 20.0 \, \text{J}$

Thus, the work done by the force is 20.0 joules.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the work done change if the peak force increased?
2. What if the displacement was increased but the force remained the same?
3. How does this calculation apply to non-linear force-displacement relationships?
4. What is the significance of the work done when the object starts and ends at rest?
5. How would you calculate work if the graph had multiple shapes?

Tip: The area under a force-displacement graph always represents the work done by the force over that displacement.