Conservative Forces

Suppose, again, that a particle is acted upon by a force ${\bf f}({\bf r})$ that is a function of the particle's displacement, ${\bf r}$. Suppose that the body travels from point $A$ to point $B$ along some particular path, labelled $1$. The net work done on the particle is

$$W_1 = \left(\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\,\cdot d{\bf r}\right)_{\rm path\,1},$$ (1.36)

where $d{\bf r}$ is an element of the path. (See Section A.14.) Suppose, now, that the particle travels between the same two points along a different path, labelled $2$. The net work done on the particle is

$$W_2 = \left(\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\,\cdot d{\bf r}\right)_{\rm path\,2}.$$ (1.37)

There are two types of forces in the universe. Conservative forces are such that

$$W_1 = W_2$$ (1.38)

irrespective of the locations of points $A$ and $B$, and the nature of paths 1 and 2. (See Section A.18.) In other words, a conservative force is such that the net work done on a particle moving between two points is independent of the path taken between the two points. Gravity is an example of a conservative force. On the other hand, non-conservative forces are such that net work done on a particle moving between two points depends on the path taken between the two points. Friction is an example of a non-conservative force.

Suppose that the particle is acted on by a conservative force and moves from point $A$ to point $B$ along path 1, and then from point $B$ to point $A$ along path 2. In other words, the particle moves in a closed loop. The net work done on the particle is

$$W = \left(\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\,\cdot d{\bf r}\right... ...+\left(\int_{{\bf r}_B}^{{\bf r}_A}{\bf f}\,\cdot d{\bf r}\right)_{\rm path\,2}$$

$$= \left(\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\,\cdot d{\bf r}\right... ...-\left(\int_{{\bf r}_A}^{{\bf r}_B}{\bf f}\,\cdot d{\bf r}\right)_{\rm path\,2}$$

$$= W_1-W_2 = 0,$$ (1.39)

where use has been made of the previous three equations. Thus, we conclude that

$$\oint {\bf f}\cdot d{\bf r} = 0.$$ (1.40)

(See Section A.18.) In other words, if a particle subject to a conservative force moves in a closed loop then zero net work is done on the particle.