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INFORMAL DEFINITION Informally, a conservative force can be thought of as a force that ''conserves'' mechanical energy. Say a particle is moving in the positive x direction with some initial Kinetic Energy Ki. Let a constant conservative force act on the particle in the negative x direction to decelerate it. Eventually, the particle will come to rest (Its kinetic energy will be zero) after a certain displacement. As the force continues to act, the particle then accelerates (in the negative direction) and returns back to its starting point. Since the only force acting on the particle as it travels this ''closed path'' is a conservative one, the particle will have the same kinetic energy Ki as it had initially, and thus the net work done by the force through this path is zero. The situation just described is known as the ''closed path test''. Any force that passes the closed path test is classified as a conservative force. The Gravitational Force , Spring Force and Electric Force (at least in a time-independent magnetic field, see Faraday's Law Of Induction for details) are examples of conservative forces, while Air Drag is the classical exemple of a non-conservative force (the energy is "given" to the air as heat and cannot be retrieved). The Magnetic Force , being orthogonal to the speed, never changes the energy of the particle it is applied on, although it depends of its speed and cannot be classified as a conservative force (it is not the gradient of a potential). PATH INDEPENDENCE is conservative.]] 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. 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 the path 1 from A to B then the path 2 backwards from B to A is 0, so the work is the same in path 1 and 2 : 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. MATHEMATICAL DESCRIPTION A force ''F'' is called ''conservative'' if it meets any of these (equivalent - Proof ) conditions:
:: abla imes ec{F} = 0. \,
::
:: abla \Phi. \, Conservative force fields are curl-less as a direct consequence of Helmholtz Decomposition . The term ''conservative force'' comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are Gravity , the electric force, and spring force. Nonconservative forces Nonconservative forces arise due to neglected Degrees Of Freedom . For instance, friction may be treated without resorting to the use of nonconservative forces by considering the motion of individual molecules; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom. Examples of nonconservative forces are Friction and non-elastic material Stress . SEE ALSO |
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