What is Work Done? Definition, Equations, & Explanation

In summary, work is a measure of change of energy and is equal to the integral of the dot product of force and displacement. The SI unit for work is the joule (J) and for power is the watt (W). Work can also be described in terms of conservative forces and potential energy, and can be used to determine gearing in a system. The work-energy theorem states that the change in work is equal to the change in kinetic energy. There is also a relativistic version of this theorem.
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Definition/Summary

Work is a measure of change of energy.

The net work done equals the change in Kinetic Energy (this is the Work-Energy Theorem).

Work is the integral of the scalar product (dot-product) of two vectors: Force and Displacement. "Displacement" means the change in position of the point at which the force is applied.

So Work is a scalar (an ordinary number), with dimensions of mass times distance-squared over time-squared.

The SI unit is the amount of Work done by a Force of one Newton acting over a displacement of one metre, and is called the joule (J), or Newton-metre (N-m).

The SI unit of Power, which is the rate of Work done, is one joule per second, and is called the watt (W).

Equations

Work is the integral of the dot product of force and displacement.

[tex]W\,=\,\int_{\mathbf{a}}^{\mathbf{a}+\mathbf{d}} \mathbf{F} \cdot d\mathbf{r}[/tex]

For a constant force, work is the dot product of the force with the total displacement.

[tex]W\,=\,\mathbf{F}\cdot\mathbf{d}\;.[/tex]

The above work equals the magnitude of the force times the magnitude of the displacement times the cosine of the angle between the force and displacement:

[tex]W\,=\,Fd\cos(\theta)\;.[/tex]


In a uniform gravitational field, Work done by gravity on a body moving along any path C starting at a height [itex]h_1[/itex] and ending at a height [itex]h_2[/itex] is:

[tex]W\,=\,\int_{(x,y,h_1)}^{(x',y',h_2)} (-mg\hat z)\cdot d\mathbf{r}\,=\,mg(h_1 - h_2)[/tex]

In an inverse-square gravitational field, Work done by gravity on a body moving along any path C starting at a height [itex]r_1[/itex] and ending at a height [itex]r_2[/itex] is:

[tex]W\,=\,\int_{r_1}^{r_2} -GmM\frac{\mathbf{r}}{r^3}\cdot d\mathbf{r}\,=\,GmM\left(\frac{1}{r_2} - \frac{1}{r_1}\right)[/tex]

which, if [itex]r_1[/itex] is very close to [itex]r_2[/itex], is approximately the same as the previous formula, with [tex]g\,=\,\frac{GM}{r_1^2}[/tex]

Extended explanation

Conservative Force:

If the work done after a total displacement of zero is zero (so the change in Kinetic Energy is zero), then the force is said to be conservative (for example, friction is not conservative, because a body moving in a full circle under friction loses Kinetic Energy, but gravity is conservative).

For a completely non-conservative force, work equals loss of mechanical energy (the energy lost generally becomes radiation or heat).

Potential energy:

For a conservative force, work done depends only on position and not on the path taken.

Potential energy is another name for work done by a conservative force.

Potential energy depends only on position, and is the work done relative to some arbitrarily-chosen position (the position of zero potential energy, chosen so as to make calculations easy).

For example, in a uniform gravitational field of strength g, when a mass m is moved by any path through a height h, the work done is mgh.

Gearing:

A machine has gearing G if the force out is G times the force in: [itex]F_1\,=\,G F_0[/itex]

If no energy is lost, then the work out equals the work in.

Since work equals force times displacement (strictly, the inner product of force and displacement), that means that the displacement of the point of application of the force out is 1/G times the displacement of the point of application of the force in: [tex]d_1\,=\,\frac{d_0}{G}[/tex]

Conversely, if a system has [itex]d_1\,\neq\,d_0[/itex], then [tex]F_1\,=\,\frac{d_0}{d_1} F_0[/tex]

For example, the gearing of a lever is the ratio of the lengths of its two "lever arms".

So a lever, or a pulley system, in which the displacement out is less than the displacement in, can lift a heavy object with a force less than its weight. :biggrin:

Derivation of Work-Energy Theorem:

[tex]\Delta\,W\ =\ \int d\mathbf{x} \cdot \mathbf{F}_{\rm net}\ =\ \int (\mathbf{v}\,dt) \cdot \left( \frac{d(m\mathbf{v})}{dt} \right)\ =\ \int d \left( \frac{1}{2} m\mathbf{v}^2 \right) \ =\ \Delta\,KE[/tex]

Relativistic version:

[tex]\Delta\,W\ =\ \int d\mathbf{x} \cdot \mathbf{F}_{\rm net}\ =\ \int (\mathbf{v}\,dt)\cdot\left(\frac{d(m\mathbf{v}/\sqrt{1
-
\mathbf{v}^2/c^2})}{dt}\right)[/tex]

[tex]=\ \int d\left(mc^2\sqrt{1
-
\mathbf{v}^2/c^2}\right)\ \ +\ \int d\left(\frac{m\mathbf{v}^2}{\sqrt{1
-
\mathbf{v}^2/c^2}}\right)\ =\ \int d\left(\frac{mc^2}{\sqrt{1
-
\mathbf{v}^2/c^2}}\right)\ =\ \Delta\,E[/tex]

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 
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Greg Bernhardt said:
Potential energy is another name for work done by a conservative force.
I disagree with this because it implies that the two are one and the same. The definition is that potential energy is the negative of the work done by a conservative force.
 
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What is the definition of work done?

Work done is the measure of the energy transferred to or from an object as a result of a force acting on it. It is calculated as the product of the force applied and the distance over which the force is applied.

What are the equations used to calculate work done?

The two main equations used to calculate work done are W = Fd (where W is work, F is force, and d is distance) and W = (mv^2)/2 (where W is work, m is mass, and v is velocity).

How is work done related to energy?

Work done is directly related to energy, as it is a measure of the energy transferred to or from an object. The unit for work, joules (J), is also the unit for energy.

What are some examples of work done?

Examples of work done include lifting objects, pushing or pulling objects, and using machines or tools to move objects. In all of these cases, a force is applied to an object over a distance, resulting in work being done.

How is work done different from power?

Work done is the measure of energy transferred, while power is the rate at which work is done. In other words, power measures how quickly work is done, while work measures the total amount of energy transferred. The unit for power is watts (W), which is equal to one joule per second.

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