The work done by a variable force can be determined by calculating the area under the force-distance graph.

To elaborate, when the force is constant, calculating the work done is straightforward: it is simply the product of the force and the distance over which it acts. Mathematically, this is expressed as:

$W = F \cdot d$

where $W$ is the work done, $F$ is the constant force, and $d$ is the distance. However, when the force varies, the situation becomes more complex. In such cases, the work done by the force is represented by the integral of the force with respect to distance. This integral effectively sums all the infinitesimally small amounts of work done by the force at each point along its path:

$W = \int F \, dx$

In practical terms, this can be visualized as calculating the area under the curve on a graph where force is plotted on the y-axis and distance on the x-axis. The area beneath this curve corresponds to the work done by the force. This relationship holds because the area of a rectangle (or any shape) on this graph is calculated as height (force) times width (distance), which aligns with the definition of work.

If the force varies linearly with distance, the graph will depict a straight line, and the area under the curve will form a triangle or trapezium, making it relatively easy to compute. Conversely, if the force varies in a more complex manner, the graph will present a curve, and calculating the area beneath it may necessitate the use of calculus.

In the context of IB Physics, you may be required to calculate the work done by a variable force either by integrating a mathematical function or by estimating the area under a curve on a graph. Both methods are grounded in the same principle: the work done by a force is the integral of that force with respect to distance.