How do you determine the work done by a variable force?

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

In more detail, when a force is constant, calculating the work done is straightforward - it's simply the force multiplied by the distance over which it acts. However, when a force varies, the situation becomes more complex. In such cases, the work done by the force is given by the integral of the force with respect to distance. This is a mathematical operation that essentially adds up all the infinitesimally small amounts of work done by the force at each point along the path.

In practical terms, this can be visualised as calculating the area under the curve on a graph of force against distance. If the force is plotted on the y-axis and distance on the x-axis, then the area under the curve represents the work done by the force. This is because the area of a rectangle (or indeed any shape) on this graph is given by height (force) times width (distance), which is the definition of work.

If the force varies linearly with distance, then the graph will be a straight line and the area under the curve will be a triangle or trapezium, which are easy to calculate. If the force varies in a more complex way, then the graph will be a curve and calculating the area under it may require calculus.

In an IB Physics context, you may be asked 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 of these methods are based on the same principle: that the work done by a force is the integral of that force with respect to distance.