Editing Work (physics) (section)

==Work and potential energy==
The scalar product of a force {{math|'''F'''}} and the velocity {{math|'''v'''}} of its point of application defines the [[power (physics)|power]] input to a system at an instant of time.  Integration of this power over the trajectory of the point of application, {{math|1=''C'' = '''x'''(''t'')}}, defines the work input to the system by the force.

===Path dependence===
Therefore, the [[mechanical work|work]] done by a force {{math|'''F'''}} on an object that travels along a curve {{math|''C''}} is given by the [[line integral]]:
<math display="block"> W = \int_C \mathbf{F} \cdot d\mathbf{x} = \int_{t_1}^{t_2}\mathbf{F}\cdot \mathbf{v}dt,</math>
where {{math|''dx''(''t'')}} defines the trajectory {{math|''C''}} and {{math|'''v'''}} is the velocity along this trajectory.
In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent.

The time derivative of the integral for work yields the instantaneous power,
<math display="block">\frac{dW}{dt} = P(t) = \mathbf{F}\cdot \mathbf{v} .</math>

===Path independence===
If the work for an applied force is independent of the path, then the work done by the force, by the [[gradient theorem]], defines a potential function which is evaluated at the start and end of the trajectory of the point of application. This means that there is a potential function {{math|''U''('''x''')}}, that can be evaluated at the two points {{math|'''x'''(''t''<sub>1</sub>)}} and {{math|'''x'''(''t''<sub>2</sub>)}} to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
<math display="block"> W = \int_C \mathbf{F} \cdot d\mathbf{x} = \int_{\mathbf{x}(t_1)}^{\mathbf{x}(t_2)} \mathbf{F} \cdot d\mathbf{x} = U(\mathbf{x}(t_1))-U(\mathbf{x}(t_2)).</math>

The function {{math|''U''('''x''')}} is called the [[potential energy]] associated with the applied force.  The force derived from such a potential function is said to be [[Conservative force|conservative]]. Examples of forces that have potential energies are gravity and spring forces.

In this case, the [[gradient]] of work yields
<math display="block" qid=Q11402> \nabla W = -\nabla U = -\left(\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}\right) = \mathbf{F},</math>
and the force '''F''' is said to be "derivable from a potential."<ref>{{Cite book|last=Taylor|first=John R.|url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PA117|title=Classical Mechanics|date=2005|publisher=University Science Books|isbn=978-1-891389-22-1|language=en}}</ref>

Because the potential {{mvar|U}} defines a force {{math|'''F'''}} at every point {{math|'''x'''}} in space, the set of forces is called a [[force field (physics)|force field]].  The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity {{math|'''V'''}} of the body, that is
<math display="block" qid=Q25342>P(t) = -\nabla U \cdot \mathbf{v} = \mathbf{F}\cdot\mathbf{v}.</math>

===Work by gravity===
[[File:Work of gravity F dot d equals mgh.JPG|right|thumb|Gravity {{math|1=''F'' = ''mg''}} does work {{math|1=''W'' = ''mgh''}} along any descending path]]
In the absence of other forces, gravity results in a constant downward acceleration of every freely moving object. Near Earth's surface the acceleration due to gravity is {{math|1=''g'' = 9.8&nbsp;m⋅s<sup>−2</sup>}} and the gravitational force on an object of mass ''m'' is {{math|1='''F'''<sub>g</sub> = ''mg''}}. It is convenient to imagine this gravitational force concentrated at the [[center of mass]] of the object.

If an object with weight {{math|''mg''}} is displaced upwards or downwards a vertical distance {{math|''y''<sub>2</sub> − ''y''<sub>1</sub>}}, the work {{math|''W''}} done on the object is:
<math display="block">W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y</math>
where ''F<sub>g</sub>'' is weight (pounds in imperial units, and newtons in SI units), and Δ''y'' is the change in height ''y''. Notice that the work done by gravity depends only on the vertical movement of the object. The presence of friction does not affect the work done on the object by its weight.

====Gravity in 3D space====
The force of gravity exerted by a mass {{mvar|M}} on another mass {{mvar|m}} is given by
<math display="block"> \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r},</math>
where {{math|'''r'''}} is the position vector from {{mvar|M}} to {{mvar|m}} and {{math|'''r̂'''}} is the unit vector in the direction of {{math|'''r'''}}.

Let the mass {{mvar|m}} move at the velocity {{math|'''v'''}}; then the work of gravity on this mass as it moves from position {{math|'''r'''(''t''<sub>1</sub>)}} to {{math|'''r'''(''t''<sub>2</sub>)}} is given by
<math display="block"> W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt.</math>
Notice that the position and velocity of the mass {{mvar|m}} are given by
<math display="block"> \mathbf{r} = r\mathbf{e}_r, \qquad\mathbf{v} = \frac{d\mathbf{r}}{dt} = \dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_t,</math>
where {{math|'''e'''<sub>''r''</sub>}} and {{math|'''e'''<sub>''t''</sub>}} are the radial and tangential unit vectors directed relative to the vector from {{mvar|M}} to {{mvar|m}}, and we use the fact that <math> d \mathbf{e}_r / dt = \dot{\theta}\mathbf{e}_t. </math>  Use this to simplify the formula for work of gravity to,
<math display="block"> W = -\int^{t_2}_{t_1}\frac{GmM}{r^3}(r\mathbf{e}_r) \cdot \left(\dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_t\right) dt = -\int^{t_2}_{t_1}\frac{GmM}{r^3}r\dot{r}dt = \frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}.</math>
This calculation uses the fact that
<math display="block"> \frac{d}{dt}r^{-1} = -r^{-2}\dot{r} = -\frac{\dot{r}}{r^2}.</math>
The function
<math display="block"> U = -\frac{GMm}{r}, </math>
is the gravitational potential function, also known as [[gravitational potential energy]].  The negative sign follows the convention that work is gained from a loss of potential energy.

===Work by a spring===
[[File:Analogie ressorts contrainte.svg|upright|right|thumb|Forces in springs assembled in parallel]]

Consider a spring that exerts a horizontal force {{math|1='''F''' = (−''kx'', 0, 0)}} that is proportional to its deflection in the ''x'' direction independent of how a body moves.  The work of this spring on a body moving along the space with the curve {{math|1='''X'''(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t''))}}, is calculated using its velocity, {{math|1='''v''' = (''v''<sub>x</sub>, ''v''<sub>y</sub>, ''v''<sub>z</sub>)}}, to obtain
<math display="block"> W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. </math>
For convenience, consider contact with the spring occurs at {{math|1=''t'' = 0}}, then the integral of the product of the distance {{mvar|x}} and the x-velocity, {{math|''xv''<sub>x</sub>''dt''}}, over time {{mvar|t}} is {{math|{{sfrac|1|2}}''x''<sup>2</sup>}}. The work is the product of the distance times the spring force, which is also dependent on distance; hence the {{math|''x''<sup>2</sup>}} result.

===Work by a gas===
The work <math>W</math> done by a body of gas on its surroundings is:
<math display="block"> W = \int_a^b P \, dV </math>
where {{mvar|P}} is pressure, {{mvar|V}} is volume, and {{mvar|a}} and {{mvar|b}} are initial and final volumes.