Editing Work (physics) (section)

==Work of forces acting on a rigid body==
The work of forces acting at various points on a single rigid body can be calculated from the work of a [[resultant force|resultant force and torque]].  To see this, let the forces '''F'''<sub>1</sub>, '''F'''<sub>2</sub>, ..., '''F'''<sub>n</sub> act on the points '''X'''<sub>1</sub>, '''X'''<sub>2</sub>, ..., '''X'''<sub>''n''</sub> in a rigid body.

The trajectories of '''X'''<sub>''i''</sub>, ''i'' = 1, ..., ''n'' are defined by the movement of the rigid body.  This movement is given by the set of rotations [''A''(''t'')] and the trajectory '''d'''(''t'') of a reference point in the body.  Let the coordinates '''x'''<sub>''i''</sub> ''i'' = 1, ..., ''n'' define these points in the moving rigid body's [[Cartesian coordinate system|reference frame]] ''M'', so that the trajectories traced in the fixed frame ''F'' are given by
<math display="block"> \mathbf{X}_i(t)= [A(t)]\mathbf{x}_i + \mathbf{d}(t)\quad i=1,\ldots, n. </math>

The velocity of the points {{math|'''X'''<sub>''i''</sub>}} along their trajectories are
<math display="block">\mathbf{V}_i = \boldsymbol{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \dot{\mathbf{d}},</math>
where {{math|'''''ω'''''}} is the angular velocity vector obtained from the skew symmetric matrix
<math display="block"> [\Omega] = \dot{A}A^\mathsf{T},</math>
known as the angular velocity matrix.

The small amount of work by the forces over the small displacements {{math|''δ'''''r'''<sub>''i''</sub>}} can be determined by approximating the displacement by {{math|1=''δ'''''r''' = '''v'''''δt''}} so
<math display="block"> \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \ldots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t</math>
or
<math display="block"> \delta W = \sum_{i=1}^n \mathbf{F}_i\cdot (\boldsymbol{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \dot{\mathbf{d}})\delta t. </math>

This formula can be rewritten to obtain
<math display="block"> \delta W = \left(\sum_{i=1}^n \mathbf{F}_i\right)\cdot\dot{\mathbf{d}}\delta t + \left(\sum_{i=1}^n \left(\mathbf{X}_i-\mathbf{d}\right)\times\mathbf{F}_i\right) \cdot \boldsymbol{\omega}\delta t = \left(\mathbf{F}\cdot\dot{\mathbf{d}}+ \mathbf{T} \cdot \boldsymbol{\omega}\right)\delta t, </math>
where '''F''' and '''T''' are the [[resultant force|resultant force and torque]] applied at the reference point '''d''' of the moving frame ''M'' in the rigid body.