Conservative and Nonconservative Forces
Conservative force—a force with the property that the work done in moving a particle between two points is independent of the path it takes.
Learning Objective

Describe properties of conservative and nonconservative forces
Key Points
 If a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.
 Conservative force is dependent only on the position of the object. If a force is conservative, it is possible to assign a numerical value for the potential at any point.
 Nonconservative force transfer the energy from the system in an energy form which can not be used by the force to transfer back to the object in motion.
Term

potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
Full Text
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.
A conservative force is dependent only on the position of the object. If a force is conservative, it is possible to assign a numerical value for the potential at any point. When an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken. Gravity and spring forces are examples of conservative forces.
If a force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points. Nonconservative forces transfer energy from the object in motion (just like conservative force), but they do not transfer this energy back to the potential energy of the system to regain it during reverse motion. Instead, they transfer the energy from the system in an energy form which can not be used by the force to transfer it back to the object in motion. Friction is one such nonconservative force.
Path Independence of Conservative Force
Work done by the gravity in a closed path motion is zero. We can extend this observation to other conservative force systems as well. We imagine a closed path motion. We imagine this closed path motion be divided in two motions between points A and B as diagramed in Fig 1 . Starting from point A to point B and then ending at point A via two work paths named 1 and 2 in the figure. The total work by the conservative force for the round trip is zero:
Motion Along Different Paths
Motion along different paths. For a conservative force, work done via different path is the same.
W=W_{AB1}+W_{BA2}=0.
Let us now change the path for motion from A to B by another path, shown as path 3. Again, the total work by the conservative force for the round trip via new route is zero : W=W_{AB3}+W_{BA2}=0.
Comparing two equations, W_{AB1}=W_{AB3}. This is true for an arbitrary path. Therefore, work done for motion from A to B by conservative force along any paths are equal.
Key Term Reference
 conservative force
 Appears in these related concepts: Gravity, Springs, and Fundamental Theorem for Line Integrals
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 friction
 Appears in these related concepts: Friction: Static, ProblemSolving With Friction and Inclines, and Inelastic Collisions in Multiple Dimensions
 gravity
 Appears in these related concepts: Defining Graviational Potential Energy, Key Points: Range, Symmetry, Maximum Height, and Properties of Electric Charges
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 position
 Appears in these related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 potential difference
 Appears in these related concepts: Energy Conservation, The ElectronVolt, and Principles of Electricity
 potential energy
 Appears in these related concepts: Problem Solving With the Conservation of Energy, Escape Speed, and Types of Energy
 scalar
 Appears in these related concepts: VectorValued Functions, Superposition of Electric Potential, and Addition and Subtraction; Scalar Multiplication
 work
 Appears in these related concepts: Heat and Work, Free Energy and Work, and The First Law of Thermodynamics
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