When Does Work Equal Force Times Distance?

In the example of the tractor pulling the plow the work did not equal Fd. The purpose of this section is to explain more fully how the quantity Fd can and cannot be used. To simplify things, I write Fd throughout this section, but more generally everything said here would be true for the area under the graph of F_II versus d.

The following two theorems allow most of the ambiguity to be cleared up.

The work-kinetic-energy theorem

The change in kinetic energy associated with the motion of an object's center of mass is related to the total force acting on it and to the distance traveled by its center of mass according to the equation ΔKE_cm = F_totald_cm.

This can be proved based on Newton's second law and the equation KE = (1/2)mv². Note that despite the traditional name, it does not necessarily tell the amount of work done, since the forces acting on the object could be changing other types of energy besides the KE associated with its center of mass motion.

The second theorem

The second theorem does relate directly to work: When a contact force acts between two objects and the two surfaces do not slip past each other, the work done equals Fd, where d is the distance traveled by the point of contact. This one has no generally accepted name, so we refer to it simply as the second theorem.

A great number of physical situations can be analyzed with these two theorems, and often it is advantageous to apply both of them to the same situation.

An ice skater pushing off from a wall.

Absorbing an impact without recoiling?

Dragging a refrigerator at constant velocity.

Accelerating a cart.