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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
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Work-Kinetic Energy Theorem as Postulate
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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.
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Definition of Work
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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:
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| Wiki Markup |
{excerpt}An [interaction] which produces a change in the [mechanical energy] of a [system], or the integrated [scalar product] of [force] and [displacement].{excerpt} h4. 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. h4. Mathematical Definition in terms of Force h6. 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]. h6. 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
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leads
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us
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to
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define
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the
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work
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done
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by
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an
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individual
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as:
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}\begin{large}\[ W = \int_{\rm path}\vec{F}\cdot d\vec{r}\]\end{large}{latex} |
The
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net
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work
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done
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on
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the
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system
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is
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then
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simply
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the
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scalar
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sum
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of
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all
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the
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individual
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works.
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Importance
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of
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Path
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Conservative
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Forces
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treated
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as
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Potential
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Energy
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The
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form
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of
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our
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definition
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of
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work
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involves
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a
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path
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integral.
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For
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some
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,
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however,
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the
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value
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of
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the
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path
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integral
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is
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determined
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solely
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by
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its
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endpoints.
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Such
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forces
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are,
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by
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definition,
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...
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.
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This
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path-independence
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is
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the
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property
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which
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allows
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us
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to
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consistently
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define
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a
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...
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to
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associate
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with
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the
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force.
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Thus,
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the
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work
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done
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by
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conservative
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forces
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will
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usually
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be
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ignored,
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since
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their
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interaction
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is
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instead
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expressed
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as
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a
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contribution
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to
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the
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...
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of
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the
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system.
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The
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two
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commonly
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considered
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conservative
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forces
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in
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introductory
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mechanics
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are:
- gravity
- elastic forces (particularly spring forces)
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Non-Conservative Forces
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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:
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* [*gravity*|gravitation (universal)] * [*elastic forces*|Hooke's Law for elastic interactions] (particularly spring forces) h6. 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. h4. 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} h4. 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. |
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.