3.2 Work Done By a Varying Force

As work is the product of force and displacement, it can be represented as the area under a graph of force as a function of displacement. In Figure 3.1, the shaded region represents the work done on an object that undergoes a constant force. Thus, the work done (area of the shaded region) can be calculated by multiplying displacement (base) by force (height).

This graphical method of calculating work is also useful in estimating the work that results from a varying force. The force applied to the object in Figure 3.2 changes over time.

Thus, the work done cannot be calculated by finding the area of a simple rectangle. However, the work can be estimated by dividing the area into small segments, calculating the area of each segment, and adding all of the segment areas.

Such areas are often divided into rectangles because the area of a rectangle is easily calculated. However, triangles, trapezoids, or any type of segments may be used. The better the segments fit the area, the more precise the estimate.

Readers with a foundation in calculus may recognize work as an integral. Indeed, Equation 3.3 presents the general form of Equations 3.1 and 3.2:

Power is the rate at which work is done. The SI unit of power, the watt (W), is the power needed to perform one joule of work per second. Notice that work, a physical quantity, is represented by a lower-case w, while watts, a unit of measure, are represented by an upper-case W.