When we are told that a person pushes on an object with a certain force, we only know how hard the person pushes: we don’t know what the pushing accomplishes. Work, W, a scalar quantity that measures the product of the force exerted on an object and the resulting displacement of that object, is a measure of what an applied force accomplishes. The harder you push an object, and the farther that object travels, the more work you have done. In general, we say that work is done by a force, or by the object or person exerting the force, on the object on which the force is acting. Most simply, work is the product of force times displacement. However, as you may have remarked, both force and displacement are vector quantities, and so the direction of these vectors comes into play when calculating the work done by a given force. Work is measured in units of joules (J), where 1 J = 1 N · m = 1 kg · m²/s².

When the force exerted on an object is in the same direction as the displacement of the object, calculating work is a simple matter of multiplication. Suppose you exert a force of 10 N on a box in the northward direction, and the box moves 5 m to the north. The work you have done on the box is N · m = 50 J. If force and displacement are parallel to one another, then the work done by a force is simply the product of the magnitude of the force and the magnitude of the displacement.

Unfortunately, matters aren’t quite as simple as scalar multiplication when the force and displacement vectors aren’t parallel. In such a case, we define work as the product of the displacement of a body and the component of the force in the direction of that displacement. For instance, suppose you push a box with a force F along the floor for a distance s, but rather than pushing it directly forward, you push on it at a downward angle of 45º. The work you do on the box is not equal to , the magnitude of the force times the magnitude of the displacement. Rather, it is equal to , the magnitude of the force exerted in the direction of the displacement times the magnitude of the displacement.

Some simple trigonometry shows us that , where is the angle between the F vector and the s vector. With this in mind, we can express a general formula for the work done by a force, which applies to all cases:

This formula also applies to the cases where F and s are parallel, since in those cases, , and , so W = F_s.

What the formula above amounts to is that work is the dot product of the force vector and the displacement vector. As we recall, the dot product of two vectors is the product of the magnitudes of the two vectors multiplied by the cosine of the angle between the two vectors. So the most general vector definition of work is:

The concept of work is actually quite straightforward, as you’ll see with a little practice. You just need to bear a few simple points in mind:

Because of the way work is defined in physics, there are a number of cases that go against our everyday intuition. Work is not done whenever a force is exerted, and there are certain cases in which we might think that a great deal of work is being done, but in fact no work is done at all. Let’s look at some examples that might be tested on SAT II Physics:

Since the gravitational force of –mg is in the same direction as the water balloon’s displacement, –h, the work done by the gravitational force on the ball is the force times the displacement, or W = mgh, where g = –9.8 m/s².

The gravitational force exerted on the balloon is still –mg, but the displacement is different. The balloon has a displacement of –h in the y direction and d (see the figure below) in the x direction. But, as we recall, the work done on the balloon by gravity is not simply the product of the magnitudes of the force and the displacement. We have to multiply the force by the component of the displacement that is parallel to the force. The force is directed downward, and the component of the displacement that is directed downward is –h. As a result, we find that the work done by gravity is mgh, just as before.

The work done by the force of gravity is the same if the object falls straight down or if it makes a wide parabola and lands 100 m to the east. This is because the force of gravity does no work when an object is transported horizontally, because the force of gravity is perpendicular to the horizontal component of displacement.

There’s a good chance SAT II Physics may test your understanding of work by asking you to interpret a graph. This graph will most likely be a force vs. position graph, though there’s a chance it may be a graph of vs. position. Don’t let the appearance of trigonometry scare you: the principle of reading graphs is the same in both cases. In the latter case, you’ll be dealing with a graphic representation of a force that isn’t acting parallel to the displacement, but the graph will have already taken this into account. Bottom line: all graphs dealing with work will operate according to the same easy principles. The most important thing that you need to remember about these graphs is:

If you recall your kinematics graphs, this is exactly what you would do to read velocity on an acceleration vs. time graph, or displacement on a velocity vs. time graph. In fact, whenever you want a quantity that is the product of the quantity measured by the y-axis and the quantity measured by the x-axis, you can simply calculate the area between the graph and the x-axis.

The work done on the box is equal to the area of the shaded region in the figure above, or the area of a rectangle of width 2 and height 4 plus the area of a right triangle of base 2 and height 2. Determining the amount of work done is simply a matter of calculating the area of the rectangle and the area of the triangle, and adding these two areas together:

If SAT II Physics throws you a curved force vs. position graph, don’t panic. You won’t be asked to calculate the work done, because you can’t do that without using calculus. Most likely, you’ll be asked to estimate the area beneath the curve for two intervals, and to select the interval in which the most, or least, work was done. In the figure below, more work was done between x = 6 and x = 8 than between x = 2 and x = 4, because the area between the graph and the x-axis is larger for the interval between x = 6 and x = 8.