6.4

Conservative Versus Nonconservative Forces

The gravitational force has an interesting property that when an object is moved from one place to another, the work done by the gravitational force does not depend on the choice of path. In Figure 6-11, for instance, an object moves from an initial height h0 to a final height hf along two different paths. As Section 6.3 discusses, the work done by gravity depends only on the initial and final heights, and not on the path between these heights. For this reason, the gravitational force is called a conservative force, according to version 1 of the following definition:

| Definition of a Conservative Force
| Version 1 A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions. Version 2 A force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point.

Figure 6-14 helps us to illustrate version 2 of the definition of a conservative force. The picture shows a roller coaster car racing through dips and double dips, ultimately returning to its starting point. This kind of path, which begins and ends at the same place, is called a closed path. Gravity provides the only force that does work on the car, assuming that there is no friction or air resistance. Of course, the track exerts a normal force, but this force is always directed perpendicular to the motion and does no work. On the downward parts of the trip, the gravitational force does positive work, increasing the car’s kinetic energy. Conversely, on the upward parts of the motion, the gravitational force does negative work, decreasing the car’s kinetic energy. Over the entire trip, the gravitational force does as much positive work as negative work, so the net work is zero, and the car returns to its starting point with the same kinetic energy it had at the start. Therefore, consistent with version 2 of the definition of a conservative force, for a closed path.

Figure 6-14 A roller coaster track is an example of a closed path.

The gravitational force is our first example of a conservative force. Later, we will encounter others, such as the elastic force of a spring and the electrical force of electrically charged particles. With each conservative force we will associate a potential energy, as we have already done in the gravitational case (see Equation 6.5). For other conservative forces, however, the algebraic form of the potential energy will differ from that in Equation 6.5.
Not all forces are conservative. A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. The kinetic frictional force is one example of a nonconservative force. When an object slides over a surface, the kinetic frictional force points opposite to the sliding motion and does negative work. Between any two points, greater amounts of work are done over longer paths between the points, so that the work depends on the choice of path. Thus, the kinetic frictional force is nonconservative. Air resistance is another nonconservative force. The concept of potential energy is not defined for a nonconservative force.
For a closed path, the total work done by a nonconservative force is not zero as it is for a conservative force. In Figure 6-14, for instance, a frictional force would oppose the motion and slow down the car. Unlike gravity, friction would do negative work on the car throughout the entire trip, on both the up and down parts of the motion. Assuming that the car makes it back to the starting point, the car would have less kinetic energy than it had originally. Table 6-2 gives some examples of conservative and nonconservative forces.

Table 6-2 Some Conservative and Nonconservative Forces

Conservative Forces
Gravitational force (Ch. 4)
Elastic spring force (Ch. 10)
Electric force (Ch. 18, 19)
Nonconservative Forces
Static and kinetic frictional forces
Air resistance
Tension
Normal force
Propulsion force of a rocket

In normal situations, conservative forces (such as gravity) and nonconservative forces (such as friction and air resistance) act simultaneously on an object. Therefore, we write the work W done by the net external force as , where Wc is the work done by the conservative forces and Wnc is the work done by the nonconservative forces. According to the work–energy theorem, the work done by the net external force is equal to the change in the object’s kinetic energy, or . If the only conservative force acting is the gravitational force, then and the work–energy theorem becomes

The work done by the gravitational force can be moved to the right side of this equation, with the result that

| (6.6)

In terms of kinetic and potential energies, we find that

| (6.7a)

Equation 6.7a states that the net work Wnc done by all the external nonconservative forces equals the change in the object’s kinetic energy plus the change in its gravitational potential energy. It is customary to use the delta symbol to denote such changes; thus, and . With the delta notation, the work–energy theorem takes the form

| (6.7b)

In the next two sections, we will show why the form of the work–energy theorem expressed by Equations 6.7a and 6.7b is useful.

Problem solving insight The change in both the kinetic and the potential energy is always the final value minus the initial value: and .

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.