Editing Work (physics) (section)

===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.