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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.