KAHIBARO
Table of Contents
Reading Area as Work
When a force acts while an object moves, the work done can often be found directly from a graph of force versus position. This is one of the most useful visual tools in mechanics, especially when the force changes from place to place.
For motion along one line, the key idea is simple. The work done by the force between two positions is the signed area under the curve on a graph of force $F$ versus position $x$.
For a force that depends on position, the work from $x_i$ to $x_f$ is
$$
W = \int_{x_i}^{x_f} F(x)\,dx
$$
On a force-position graph, this integral is the area under the curve, counted with sign.
If the force is above the horizontal axis, the work is positive. If the force is below the axis, the work is negative. If the graph crosses the axis, the total work is the algebraic sum of the positive and negative areas.
Constant Force as a Rectangle
The simplest case is a constant force. If $F$ does not change with position, then the graph is a horizontal line. The area under it is a rectangle, so
$$
W = F \Delta x
$$
where $\Delta x = x_f - x_i$.
This matches the usual formula for work by a constant force in one dimension. On the graph, the height is the force and the width is the displacement.
Variable Force and Curved Area
In many physical situations, the force is not constant. It may increase, decrease, or change direction as the object moves. In that case, the graph is not a horizontal line, and the work is the area under the curve.
If the force changes smoothly, we imagine splitting the motion into many tiny intervals of position. Over each tiny interval, the force is almost constant, so each small contribution to work is approximately
$$
dW = F(x)\,dx
$$
Adding all these small contributions gives the integral.
Positive and Negative Work on the Graph
A force-position graph shows not just how much work is done, but also whether the force helps or opposes the motion.
If the force and displacement are in the same direction, the graph lies above the axis and the work is positive. If the force points opposite to the displacement, the graph lies below the axis and the work is negative.
For example, suppose a force is positive over one part of the path and negative over another. Then
$$
W_{\text{total}} = W_{\text{positive area}} + W_{\text{negative area}}
$$
Here, the negative area must be subtracted.
Area above the $x$ axis gives positive work.
Area below the $x$ axis gives negative work.
Total work is the signed area, not the total geometric area.
Common Geometric Shapes
Often the graph has simple shapes, so the work can be found using geometry instead of calculus. This is especially useful in introductory problems.
If the region under the graph is a rectangle, triangle, or trapezoid, use the usual area formulas.
| Shape under graph | Area formula | Work |
|---|---|---|
| Rectangle | $A = bh$ | $W = F \Delta x$ |
| Triangle | $A = \frac{1}{2}bh$ | $W = \frac{1}{2}F\Delta x$ |
| Trapezoid | $A = \frac{1}{2}(F_1+F_2)\Delta x$ | $W = \frac{1}{2}(F_1+F_2)\Delta x$ |
Here, the base $b$ is the interval in position, and the height is the force value.
Example with a Triangle
Suppose the force increases linearly from $0\ \text{N}$ at $x=0$ to $6\ \text{N}$ at $x=4\ \text{m}$. The graph is a straight line, and the area under it is a triangle.
So the work is
$$
W = \frac{1}{2} \times 4 \times 6 = 12\ \text{J}
$$
The unit is joules because
$$
\text{N}\cdot\text{m} = \text{J}
$$
Spring Force as a Graph Example
A very important example is the spring force. For a spring, the force depends on position according to Hooke's law,
$$
F(x) = -kx
$$
The meaning of this law belongs to the chapter on spring force and spring oscillations, but here we focus only on the graph. The force-position graph is a straight line through the origin with negative slope.
If an external force slowly stretches the spring from $x=0$ to $x$, the magnitude of the applied force grows linearly, and the work done by that applied force is the triangular area:
$$
W = \frac{1}{2}kx^2
$$
This result comes directly from the area of a triangle with base $x$ and height $kx$.
For a linearly increasing force from $0$ to $F_{\max}$ over a distance $x$,
$$
W = \frac{1}{2}F_{\max}x
$$
For a spring stretched from equilibrium to displacement $x$,
$$
W = \frac{1}{2}kx^2
$$
Piecewise Graphs
Sometimes the graph consists of several separate parts. For example, the force may be constant over one interval, then decrease linearly, then become negative. In such cases, find the area of each section and add them with signs.
Suppose a graph has three sections. Then the total work is
$$
W = W_1 + W_2 + W_3
$$
This method is often easier than trying to handle the whole graph at once.
What the Slope Means, and What It Does Not Mean
It is important not to confuse a force-position graph with other graphs used in mechanics.
On a force-position graph, the area under the graph gives work. The slope of the graph tells how the force changes with position, but the slope itself is not the work.
This is different from other graphs, where slope or area may represent different physical quantities. Here, the special rule is that area equals work.
On a force-position graph:
Area under the curve $\rightarrow$ work
Slope of the curve $\rightarrow$ rate at which force changes with position
Do not confuse slope with work.
Units of Work from the Graph
The vertical axis has units of force, usually newtons, and the horizontal axis has units of position, usually meters. Therefore the area has units
$$
\text{N} \cdot \text{m} = \text{J}
$$
So any area you calculate from the graph should come out in joules.
Final Idea
A force-position graph turns the calculation of work into a geometric problem. Instead of only using formulas, you can often look at the graph and find the work by computing signed area. This is especially powerful when the force varies with position.
The central rule of this chapter is
$$
W = \int_{x_i}^{x_f} F(x)\,dx
$$
For simple graphs, this means:
work = signed area under the force-position curve