# Work, Energy, and Energy Resources

## Work: The Scientific Definition


### Learning Objectives

By the end of this section, you will be able to:
1. Explain how an object must be displaced for a force on it to do work.
2. Explain how relative directions of force and displacement determine whether the work done is positive, negative, or zero.

### What It Means to Do Work

The scientific definition of work differs in some ways from its everyday meaning. Certain things we think of as hard work, such as writing an exam or carrying a heavy load on level ground, are not work as defined by a scientist. The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred.
For work, in the scientific sense, to be done, a force must be exerted and there must be displacement in the direction of the force. 
Formally, the work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as
where  is work,  is the displacement of the system, and  is the angle between the force vector  and the displacement vector , as in . We can also write this as
To find the work done on a system that undergoes motion that is not one-way or that is in two or three dimensions, we divide the motion into one-way one-dimensional segments and add up the work done over each segment.
To examine what the definition of work means, let us consider the other situations shown in . The person holding the briefcase in (b) does no work, for example. Here , so . Why is it you get tired just holding a load? The answer is that your muscles are doing work against one another, but they are doing no work on the system of interest (the “briefcase-Earth system”—see Gravitational Potential Energy for more details). There must be displacement for work to be done, and there must be a component of the force in the direction of the motion. For example, the person carrying the briefcase on level ground in (c) does no work on it, because the force is perpendicular to the motion. That is, , and so . 
In contrast, when a force exerted on the system has a component in the direction of motion, such as in (d), work is done—energy is transferred to the briefcase. Finally, in (e), energy is transferred from the briefcase to a generator. There are two good ways to interpret this energy transfer. One interpretation is that the briefcase’s weight does work on the generator, giving it energy. The other interpretation is that the generator does negative work on the briefcase, thus removing energy from it. The drawing shows the latter, with the force from the generator upward on the briefcase, and the displacement downward. This makes , and ; therefore,  is negative.

### Calculating Work

Work and energy have the same units. From the definition of work, we see that those units are force times distance. Thus, in SI units, work and energy are measured in newton-meters. A newton-meter is given the special name joule (J), and . One joule is not a large amount of energy; it would lift a small 100-gram apple a distance of about 1 meter.  

### Test Prep for AP Courses


### Section Summary

1. Work is the transfer of energy by a force acting on an object as it is displaced.
2. The work  that a force  does on an object is the product of the magnitude  of the force, times the magnitude  of the displacement, times the cosine of the angle  between them. In symbols,


3. The SI unit for work and energy is the joule (J), where .
4. The work done by a force is zero if the displacement is either zero or perpendicular to the force.
5. The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.

### Conceptual Questions


### Problems & Exercises
