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

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

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

The net work done on the system is then simply the scalar sum of all the individual works.

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}\[ W^{\rm NC}_{\rm net} = E_{f} - E_{i} \] \end{large}{latex}

h3. 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.

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