Path Independence
A direct consequence of the closed path test is that the work done by a conservative force on a particle moving between any two points does not depend on the path taken by the particle. Also the work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof of that, let's imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.
For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the top of the slide to the bottom will be the same no matter what the shape of the slide; it can be straight or it can be a spiral. The amount of work done only depends on the vertical displacement of the child.
Read more about this topic: Conservative Force
Famous quotes containing the words path and/or independence:
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—William Shakespeare (1564–1616)
“Children are as destined biologically to break away as we are, emotionally, to hold on and protect. But thinking independently comes of acting independently. It begins with a two-year-old doggedly pulling on flannel pajamas during a July heat wave and with parents accepting that the impulse is a good one. When we let go of these small tasks without anger or sorrow but with pleasure and pride we give each act of independence our blessing.”
—Cathy Rindner Tempelsman (20th century)