Work of External Forces

Whenever a force is displaced by a certain amount or a displacement is induced by a certain force, work is generated. The increment of work done on a body by a force F during an incremental displacement ds from its point of application is where  is the angle between F and ds (Fig. 3.57). Equation (3.91) implies that work is the product of force and the component of displacement in the line of action of the force, or the product of displacement and the component of force along the path of the displacement.
If the component of the displacement is in the same direction as the force or the component of the force acts in the same direction as the path of displacement, the work is positive; otherwise, the work is negative. When the line of action of the force is perpendicular to the direction of displacement , no work is done. When the displacement is a finite quantity, the total work can be expressed as

Integration is carried out over the path the force travels, which may not be a straight line.
The work done by the weight of a body, which is the force, when it is moved in a vertical direction is the product of the weight and vertical displacement. According to Eq. (3.91) and with  the angle between the downward direction of gravity and the imposed displacement, the weight does positive work when movement is down. It does negative work when movement is up.
In a similar fashion, the rotation of a body by a moment M through an incremental angle d also generates work. The increment of work done in this case is


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