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{excerpt}An [interaction] which produces a change in the [mechanical energy] of a [system], or the integrated [scalar product] of [force] and [displacement].{excerpt}

h3. 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|gravitation (universal)] 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.

h3. Mathematical Definition in terms of Force

h4.  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].

h4. Definition of Work

By comparing the derivation of the [theorem|Work-Kinetic Energy Theorem] to its statement, we see that in order for the [theorem|Work-Kinetic Energy Theorem] to be satisfied, we must make the definition:

{latex}\begin{large}\[ W_{net} = \int_{path} \vec{F}_{net}\cdot d\vec{r}\]\end{large}{latex}

which leads us to define the work done by an individual [force] as:

{latex}\begin{large}\[ W = \int_{path}\vec{F}\cdot d\vec{r}\]\end{large}{latex}

h3. Importance of Path

h4. Conservative Forces treated as Potential Energy

The form of our definition of work involves a path integral.  For some [forces|force], however, the value of the path integral is determined solely by its endpoints.  Such forces are, by definition, [conservative forces|conservative force].  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*|gravitation (universal)]
* [*elastic forces*|Hooke's Law for elastic interactions] (particularly spring forces)

h4. Non-Conservative Forces

For forces other than [gravity|gravitation (universal)] and [elastic forces|Hooke's Law for elastic interactions], 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.

h3. Mathematical Definition in terms of Mechanical Energy

If all [conservative interactions|conservative force] present within a [system] are described as [potential energies|potential energy] 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:

{latex}\begin{large}\[ W_{\rm nc} = E_{f} - E_{i} \] \end{large}{latex}



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