7.3 Work-energy theorem

We have discussed how to find the work done on a particle by the forces that act on it, but how is that work manifested in the motion of the particle? According to Newton’s second law of motion, the sum of all the forces acting on a particle, or the net force, determines the rate of change in the momentum of the particle, or its motion. Therefore, we should consider the work done by all the forces acting on a particle, or the net work    , to see what effect it has on the particle’s motion.

Let’s start by looking at the net work done on a particle as it moves over an infinitesimal displacement, which is the dot product of the net force and the displacement: $d{W}_{\text{net}}={\overrightarrow{F}}_{\text{net}}·d\overrightarrow{r}.$ Newton’s second law tells us that ${\overrightarrow{F}}_{\text{net}}=m(d\overrightarrow{v}\text{/}dt),$ , so $d{W}_{\text{net}}=m(d\overrightarrow{v}\text{/}dt)·d\overrightarrow{r}.$ For the mathematical functions describing the motion of a physical particle, we can rearrange the differentials dt , etc., as algebraic quantities in this expression, that is,

where we substituted the velocity for the time derivative of the displacement and used the commutative property of the dot product [ [link] ]. Since derivatives and integrals of scalars are probably more familiar to you at this point, we express the dot product in terms of Cartesian coordinates before we integrate between any two points A and B on the particle’s trajectory. This gives us the net work done on the particle:

In the middle step, we used the fact that the square of the velocity is the sum of the squares of its Cartesian components, and in the last step, we used the definition of the particle’s kinetic energy. This important result is called the work-energy theorem    ( [link] ).

According to this theorem, when an object slows down, its final kinetic energy is less than its initial kinetic energy, the change in its kinetic energy is negative, and so is the net work done on it. If an object speeds up, the net work done on it is positive. When calculating the net work, you must include all the forces that act on an object. If you leave out any forces that act on an object, or if you include any forces that don’t act on it, you will get a wrong result.

The importance of the work-energy theorem, and the further generalizations to which it leads, is that it makes some types of calculations much simpler to accomplish than they would be by trying to solve Newton’s second law. For example, in Newton’s Laws of Motion , we found the speed of an object sliding down a frictionless plane by solving Newton’s second law for the acceleration and using kinematic equations for constant acceleration, obtaining