# 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.

#### 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.

#### Terms

• A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.

• The vector field denoting the rotationality of a given vector field.

• The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.

#### Figures

1. ##### Motion Along Different Paths

Motion along different paths. For a conservative force, work done via different path is the same.

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 Figure 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:

W=WAB1+WBA2=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=WAB3+WBA2=0.

Comparing two equations, WAB1=WAB3. This is true for an arbitrary path. Therefore, work done for motion from A to B by conservative force along any paths are equal.

### Mathematical Description

A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions:

•  1. The curl of F is zero: <equation contenteditable="false">$\nabla \times \vec{F} = 0. \,$
• 2. There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place: $W \equiv \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$
• 3. The force can be written as the negative gradient of a potential $\Phi$ : $\vec{F} = -\nabla \Phi. \,$

#### Key Term Glossary

conservative force
A force with the property that the work done in moving a particle between two points is independent of the path taken.
##### Appears in these related concepts:
curl
The vector field denoting the rotationality of a given vector field.
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energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
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friction
A force that resists the relative motion or tendency to such motion of two bodies in contact.
##### Appears in these related concepts:
The rate at which a physical quantity increases or decreases relative to change in a given variable, especially distance.
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gravity
Resultant force on Earth's surface, of the attraction by the Earth's masses, and the centrifugal pseudo-force caused by the Earth's rotation.
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motion
A change of position with respect to time.
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particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
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position
A place or location.
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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.
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potential difference
The difference in potential energy between two points in an electric field; the difference in charge between two points in an electrical circuit; voltage.
##### Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
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scalar
A quantity that has magnitude but not direction; compare vector.
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Scalar
A quantity which can be described by a single number, as opposed to a vector which requires a direction and a number.
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trajectory
The path of a body as it travels through space.
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vector
A directed quantity, one with both magnitude and direction; the between two points.
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vector field
a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
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work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.