Path Independence of Conservative Forces

Path Independence of Conservative Forces

The primary test for determining whether a force is conservative or non conservatives 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 10 particle moving around every closed path is zero.

For example, suppose that a particle moves from point a to point b in  along either path I 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, can write this result as.

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.

(0) As a conservative force acts on it, a particle can move from point 0 to point b along either path I or path 2. (b) The particle moves in a round trip, from point 0 to point b along path I and then back to point 0 along path 2
(0) As a conservative force acts
on it, a particle can move from point 0
to point b along either path I or path 2.
(b) The particle moves in a round trip,
from point 0 to point b along path I
and then back to point 0 along path 2

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