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.
Work When Force and Displacement Are Parallel
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.
Work When Force and Displacement Are Not Parallel
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 = Fs.
Dot Product
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:
Review
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:
