Conservative and Nonconservative forces

Let us list the key elements of the two situations we just discussed:

1. The system consists of two or more objects.

2. A force acts between a particle-like object (tomato or block) in the system and the rest of the system.

3. When the system configuration changes, the force does work (call it ) on the particle-like object, transferring energy between the kinetic energy of the object and some other form of energy of the system.

4. When the configuration change is reversed, the force reverses the energy transfer, doing work in the process.

In a situation in which is always true, the other form of energy is a potential energy, and the force is said to be a conservative force. As you might suspect, the gravitational force and the spring force are both conservative (since otherwise we could not have spoken of gravitational potential energy and elastic potential energy, as we did previously).

A force that is not conservative is called a nonconservative force. The kinetic frictional force and drag force are nonconservative. For an example, let us send a block sliding across a floor that is not frictionless. During the sliding, a kinetic frictional force from the floor does negative work on the block, slowing the block by transferring energy from its kinetic energy to a form of energy called thermal energy (which has to do with the random motions of atoms and molecules). We know from experiment that this energy transfer cannot be reversed (thermal energy cannot be transferred back to kinetic energy of the block by the kinetic frictional force). Thus, although we have a system (made up of the block and the floor), a force that acts between parts of the system, and a transfer of energy by the force, the force is not conservative. Therefore, thermal energy is not a potential energy.

When only conservative forces act on a particle-like object, we can greatly simplify otherwise difficult problems involving motion of the object. The next section, in which we develop a test for identifying conservative forces, provides one means for simplifying such problems.

2 Path Independence of Conservative forces

The primary test for determining whether a force is conservative or nonconservative is this: Let the force act on a particle that moves along any closed path, beginning at some initial position and eventually returning to that position (so that the particle makes a round trip beginning and ending at the initial position). The force is con­servative only if the total energy it transfers to and from the particle during the round trip along this and any other closed path is zero. In other words:

The net work done by a conservative force on a particle moving around every closed path is zero.

We know from experiment that the gravitational force passes this closed-path test. An example is the tossed tomato of Fig. 8-2. The tomato leaves the launch point with speed and kinetic energy . The gravitational force acting on the tomato slows it, stops it, and then causes it to fall back down. When the tomato returns to the launch point, it again has speed and kinetic energy . Thus, the gravitational force transfers as much energy from the tomato during the ascent as it transfers to the tomato during the descent back to the launch point. The net work done on the tomato by the gravitational force during the round trip is zero. An important result of the closed-path test is that

► The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle.

For example, suppose that a particle moves from point a to point b in Fig. 8-4a along either path 1 or path 2. If only a conservative force acts on the particle, then the work done on the particle is the same along the two paths. In symbols, we can write this result as

(8-2)

where the subscript ab indicates the initial and final points, respectively, and the subscripts 1 and 2 indicate the path.

This result is powerful, because it allows us to simplify difficult problems when only a conservative force is involved. Suppose you need to calculate the work done by a conservative force along a given path between two points, and the calculation is difficult or even impossible without additional information. You can find the work by substituting some other path between those two points for which the calculation is easier and possible. Sample Problem 8-1 gives an example, but first we need to prove Eq. 8-2.

Date: 2015-01-12; view: 1207