Editing Work (physics) (section)

===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.