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Mainly From Wikipedia, the free encyclopedia Conservative & Non-conservative Forces Forces can be classified as conservative or non-conservative. Conservative forces are equivalent to the gradient of a potential while non-conservative forces are not. If a potential energy can be associated with a force, we call that force conservative. If a potential energy cannot be associated with a force, we call that force non-conservative. The spring force & gravitational force are conservative; the frictional force is non-conservative. Mechanical energy [K+U] is conserved only if no non-conservative forces are acting. The inverse square laws of gravitation and electrostatics are examples of central forces where the force exerted by one particle on another is along the line joining them and is also independent of direction. Whatever the variation of force with distance, a central force can always be represented by a potential; forces for which a potential can be found are called conservative. Potential energy Instead of a force, often the mathematically related concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field and opposite to the force produced at every point: is defined as that field whose gradient is equal Conservative force(s) A conservative force is defined as a force that does not depend on the path taken to increase in potential energy. Informal definition Informally, a conservative force can be thought of as a force that conserves mechanical energy. Suppose a particle starts at point A, and there is a constant force F acting on it. Then the particle is moved around by other forces, and eventually ends up at A again. Though the particle may still be moving, at that instant when it passes point A again, it has traveled a closed path. If the net work done by F at this point is 0, then F passes the closed path test. Any force that passes the closed path test is classified as a conservative force. The gravitational force, spring force, magnetic 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 friction and air drag are classical examples of non-conservative forces (the energy is transferred to the air as heat and cannot be retrieved). Path independence The work done by the gravitational force on an object depends only on its change in height because the gravitational force 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. 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. Mathematical description A force F is called conservative if it meets any of these (equivalent - proof) conditions: The curl of F is zero: The work, W, is zero for any simple closed path: The force can be written as the gradient of a potential, Φ: 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. A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space, [48] and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area. Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models which are dependent on a position often given as a radial vector emanating from spherically symmetric potentials.[49] Examples of this follow: For gravity: where G is the gravitational constant, and mn is the mass of object n. For electrostatic forces: where ε0 is electric permittivity of free space, and qn is the electric charge of object n. For spring forces: where k is the spring constant Conservative forces (continued) If the vector field associated to a force is conservative then the force is said to be a conservative force.