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Conservative Forces

What is the use of work-energy theorem unless we can calculate the integral which requires the knowledge of how the force F acts throughout the motion? It turns out that for certain type of forces, the work done depends only upon the initial and final positions (1 and 2) of the object and not on the path through which the object moves from 1 to 2. Such forces are called conservative forces.

In other words, suppose a particle, subjected to force F, is moved from position a to b by two arbitrary paths, acb and adb. If the work done by force in both the processes is same, that is, if

then the force F is called conservative. We show below that the uniform-constant forces due to electric field (q E) and gravity ( m g ) are typical examples of conservative forces. In general, a force which depends explicitly upon the velocity of the particle is not conservative. (However, the magnetic force ( q v × B ) can be included among conservative forces in the sense that it acts perpendicular to velocity and hence work done by it is always identically zero).

In particular, if a conservative force F is an explicit function of r only (and not of t), then the net work done by the force in moving a particle around any closed-path (or loop) in space is zero. To show this, let us now move the particle from b to a along any path. The work done during this process is,

which is path independent for a conservative force. Mathematically, if F = F ( r ), then we have

That is written as,

where the sign means we have taken the integral along a closed path

The forces for which the integral ∫ F . dl depends upon the path are called non-conservative forces. Velocity dependent forces are in general non-conservative. A typical example of non-conservative force is friction which always acts opposite to direction of motion. The total work done by friction depends on the path (i.e. on the distance moved).