Definition of a Conservative Force

So exactly what kinds of forces conserve mechanical energy? To answer this we consider particles traveling in closed loops under the influence of the forces in question. In other words, a closed loop describes a "round trip", during which the particle is under influence of the force. Many systems produce closed loops, such as a ball bouncing up and down, or a mass on a spring. If a conservative force acts upon the particle during this closed loop, the velocity of the particle at the beginning and the end of the loop must be the same. Why? Because if the velocity is any different, the kinetic energy of the particle will be different, meaning that mechanical energy must not have been conserved. Thus we come to our first statement about conservative forces:

If a body is under the action of a force that does no net work during any closed loop, then the force is conservative. If work is done, the force is nonconservative.

In other words, a particle located at the same physical location in a closed loop must have the same kinetic energy at all times if it is within a conservative system. This fact is the fundamental definition of a conservative force. Though we will derive other properties of conservative forces from this statement, it remains the most important one to keep in mind.

Since the work over a closed loop must be zero for conservative forces, what other properties can we state? Let's break the path of a closed loop into two separate paths:

Since the path in part a) is a closed loop, we know that the total work around the loop must be zero if the force in question is conservative: W ₁ + W ₂ = 0 . Now compare the two different paths taken from A to B in part b). The work on the first path is the same as part a), simply W ₁ . The direction of travel on the second path is reversed in b), implying that the work done over the path is negated, or equal to - W ₂ . But we know from a) that - W ₂ = W ₁ . Thus the work done over path 1 and path 2 in part b) is the same! This concept, called path independence, is incredibly useful, as we will soon see. Stated verbally:

The work done by a conservative force in moving a body from an initial location to a final location is independent of the path taken between the two points

Let's examine the implications of this statement. Consider a particle moving between two points in an odd shaped path. Our old definition of work demands that we evaluate the work done at each part of the odd path in order to evaluate the total work done over the journey, and thus the change in kinetic energy and velocity. With this just-stated principle of conservative forces, however, we can use any path we like: a straight line, a circular arc, or a path in which the work done on the particle is constant. Though our first statement about conservative forces is powerful, this second statement proves to be the most applicable: we will use this concept to solve numerous problems in the sections to come.

Examples of Conservative and Nonconservative Forces

Such abstract principles might be confusing. In order to clarify these two very important concepts, we will examine two forces: gravity, a conservative force, and friction, a nonconservative one.

Gravity

Gravity is the most common conservative force, and to demonstrate that it is conservative is relatively simple. Consider first a ball thrown up into the air. On the ball's trip upward, gravity works against the motion of the ball, producing a total work of - mgh . This negative work causes the ball to slow down until it stops, reverses direction and begins to fall. During its fall, the force of gravity is in the same direction as the motion of the ball, and the gravitational force does positive work of magnitude mgh , accelerating ball until it reaches the ground with the same speed with which it left. What is the net work done by gravity on the ball over this closed loop? Zero, as we expect by our first principle of conservative forces.

What about our second principle? Let's construct two alternative paths for a ball being thrown up into the air:

Here we have path 1, a straight vertical line from A to B, and the path consisting of segments 2,3 and 4, which has vertical and horizontal components. We expect that the work done over these two segments are equal. The work over the first path is simple. The gravitational force always opposes the motion, and exerts a net work on the ball of - mgh . The work over the second path requires three calculations, one for each line segment. On segment 2, the horizontal one, the force on the ball is always perpendicular to the motion of the ball, implying that the work done on the ball over this segment is zero. The same is true for segment 4. Segment 3 is identical to segment 1, experiencing a net work of - mgh . Since the work over segment 2 and 4 is zero, the total work over the second path is - mgh , the same as the first one. We have demonstrated path independence, and thus the conservative nature of gravity.

Friction

Friction is the most common nonconservative force, and we will demonstrate why it is not conservative. Consider a crate on a rough floor, of weight W. The crate is pushed from one end of the floor to the other, a distance of h meters, and then back to its original spot. What is the net work done on the crate? At all times the friction opposes the motion of the crate, exerting a force of μ _k W at all times. Thus the total work done over the trip is simply (- 2)(μ _k W)(h) = - 2hwμ _k , clearly not equal to zero. The net work by friction over a closed path is not zero, and it is nonconservative.

Is friction path independent? We expect not, because we know it is nonconservative. To prove the suspicion, simply consider two possible ways to move a crate between two points on a rough floor. One is a straight line, one is a somewhat longer route. No matter the path, the force is the same at all times that the crate is moving. The difference, however, is that friction acts over a longer distance in the case of the second path, causing a greater net work to be done. Thus friction is not path independent, and we confirm that it is nonconservative.

The distinctions between conservative and nonconservative forces may seem somewhat arbitrary at this point. However, in the next section we will see that conservative forces, because of the properties developed in this section, allow for incredible simplification of otherwise difficult mechanics problems.