Work-Kinetic Energy Theorem

The basic work-kinetic energy theorem applies to a system consisting of a single particle only. The general work-kinetic energy system applies to a system of particles that may interact with each other. A system of particles may be gaseous, liquid, or solid. The two theorems are described below.

Work-kinetic energy theorem for a single particle (mass point)

If a particle is subject to a net force vector Fnet, the work Wnet done by the net force on the particle during a given time interval is equal to the change in the particle's kinetic energy during that time interval. In symbols,

Wnet = DKE space(1)

where

KE = (m/2)v2 space(2)

is the particle's kinetic energy, m the mass of the particle and v the speed of the particle.

Work-kinetic energy theorem for a system of particles

Eq.(1) above also expresses the work-kinetic energy for a system of particles, except that W now is the total work done on all particles of the system and KE is the total kinetic energy of all particles of the system.

In more detail, the total work W done on a system of particles and the total kinetic energy KE of a system are defined as follows. Let the system consist of N particles, labeled 1 to N. Let us consider some time interval during which the net force vector F1 acting on Particle 1 does work W1 on Particle 1 and the kinetic energy of Particle 1 changes by DKE1. Similarly for Particles 2 to N. Then

W = W1 + W2 + ... + WN space(3)

and

KE = KE1 + KE2 + ... + KEN. space(4)

To derive the work-kinetic energy theorem for a system of particles, all one has to do is write down Eq.(1) separately for each particle and sum the resulting N equations.

Comment. When dealing with a system of particles, one distinguishes between internal and external forces acting on the particles of the system. Internal forces are forces exerted on a particle by other particles of the system, and external forces are forces exerted on a particle by objects that are not part of the system.

One can write the total work done on the system as the sum of the total work done on all particles by the internal forces, Wint, and the total work done on all particles by the external forces, Wext,

W = Wint + Wext. space(5)