Work-Energy Theorem

The work-energy theorem must be distinguished from the work-kinetic energy theorem. What adds to the confusion is that the latter is often referred to as the work-energy theorem. In MAP, the two names are used to distinguish between two different theorems.

The work-energy theorem says that

the total work done on a mechanical system during a certain time interval by all non-conservative forces acting on the system is equal to the change in the system's mechanical energy during that time interval.

Comment 1. The mechanical energy of a system is the sum of its kinetic energy and its potential energy.

Comment 2. All external conservative forces doing work on the system are to be included among the non-conservative forces in this theorem, because no potential energy is associated with external conservative forces.

The non-conservative forces acting on the system may be either external or internal forces.

Example. Let the system consist of a block, a fixed incline, and the earth supporting the incline. There is friction between the block and the incline, and the block can slide on the incline. An applied force is acting on the block.

The applied force acting on the block and the friction forces acting on the block and the incline are the non-conservative forces acting on the system. The applied force is an external non-conservative force and the friction forces are internal non-conservative forces.

The gravitational force acting on the block is an internal conservative force of the system and is taken into consideration through the gravitational potential energy of the block-earth pair of bodies. The gravitational force exerted by the block on the earth does a negligible amount of work which will be neglected.

The work-kinetic energy theorem implies that during a given time interval the sum of the work done by the applied force on the block, Wapp, and the work done by the friction force on the block, Wfric, is equal to the change in the block's mechanical energy, DE. In symbols,

Wapp + Wfric = DE, where E = KE + PE.

The friction force acting on the incline does no work because the incline is not moving. Therefore this work does not contribute to the left-hand side of the equation above.