An interaction which produces a change in the mechanical energy of a system, or the integrated scalar product of force and displacement.
Motivation for Concept
It requires effort to alter the mechanical energy of an object, as can clearly be seen when attempting to impart kinetic energy by pushing a car which has stalled or to impart gravitational potential energy by lifting a heavy barbell. We would like to quantify what we mean by "effort". It is clear that force alone is not enough to impart mechanical energy. Suppose that the car or the barbell is just too heavy to move. Then, for all the pushing or pulling that is done (a considerable force), no energy is imparted. For the mechanical energy of a system to change, the system must alter its position or its configuration. In effect, the force must impart or reduce motion in the system to which it is applied. Thus, work requires two elements: force and motion.
Mathematical Definition in terms of Force
Work-Kinetic Energy Theorem as Postulate
Suppose that we postulate the Work-Kinetic Energy Theorem for a point particle as the defining relationship of work. Doing so will allow us to find a mathematical definition of work in terms of force.
Definition of Work
By comparing the derivation of the theorem to its statement, we see that in order for the theorem to be satisfied, we must make the definition:
which leads us to define the work done by an individual force as:
The net work done on the system is then simply the scalar sum of all the individual works.
Importance of Path
Conservative Forces treated as Potential Energy
The form of our definition of work involves a path integral. For some forces, however, the value of the path integral is determined solely by its endpoints. Such forces are, by definition, conservative forces. This path-independence is the property which allows us to consistently define a potential energy to associate with the force. Thus, the work done by conservative forces will usually be ignored, since their interaction is instead expressed as a contribution to the mechanical energy of the system. The two commonly considered conservative forces in introductory mechanics are:
- gravity
- elastic forces (particularly spring forces)
Non-Conservative Forces
For forces other than gravity and elastic forces, it is usually impossible to define a useful potential energy, and so the path of the system must be understood in order to compute the work when energy is used to describe a system subject to these interactions.
Mathematical Definition in terms of Mechanical Energy
If all conservative interactions present within a system are described as potential energies then it is possible to define the net non-conservative work done on the system as a change in the mechanical energy of the system:
Negative Work
It is important to note that work is a scalar, and has no direction associated with it. Work can still be negative, however, as can be clearly seen from the definition of net work in terms of mechanical energy. Whenever the mechanical energy of a system decreases, the system has experienced a net negative work. By looking at the definition of work in terms of force, it is possible to show that an individual force will do negative work will occur whenever that force is directed against the displacement (directed more than 90° away from the direction of travel). In everyday language, negative work is produced by any force which is attempting to slow the system's movement.