7.11: Work Done Over an Inclined Plane
The center-of-mass framework helps to easily describe the work done on rigid bodies. Since the internal forces in a rigid body do no work, they can be ignored, and the external forces can be considered in the work-energy theorem.
The work done by gravity to move a rigid body, or the work done by an opposing force to move a rigid body against gravity, can be calculated using the center-of-mass framework. It is the line integral of the force of gravity over the path, considered positive if gravity increases its kinetic energy and negative if work is done against gravity to lift it, thereby reducing its kinetic energy. In the case of horizontal motion, the work done by or against gravity is zero. That is, the change in kinetic energy of a body, even if non-zero, is purely due to other forces.
The formulation holds equally when a body moves over an inclined plane. The only difference is that only a component of gravity is along the plane, and only this component needs to be considered while calculating the work done. The gravitational force normal to the plane does no work at any point during the displacement.