Conservative force

Definition and Characteristics of Conservative Forces
– A conservative force is a force in physics that has the property of the total work done in moving a particle between two points being independent of the path taken.
– It can also be defined as a force that, when a particle travels in a closed loop, the total work done by the force is zero.
– A conservative force depends only on the position of the object and can be assigned a numerical value for the potential at any point.
– Examples of conservative forces include gravitational force, force in an elastic spring, electrostatic force between two electric charges, and magnetic force between two magnetic poles.
– Non-conservative forces, such as frictional force, do not exhibit these properties.
– Informally, a conservative force can be thought of as a force that conserves mechanical energy.
– If a particle starts at a point and is moved around by other forces, eventually ending up at the starting point again, a conservative force will have a net work done of 0 at that point.
– Any force that passes this closed path test for all possible closed paths is classified as a conservative force.
– Non-conservative forces, such as friction and air drag, result in a loss of mechanical energy that is converted into other forms, like heat and sound energy.
– A 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.
– This means that the work done by a conservative force is the same for different paths that start and end at the same points.
– Mathematically, this can be proven using the concept of the curl of a force field and the negative gradient of a potential.
– The path independence property is a distinguishing characteristic of conservative forces.
– A force field is considered a conservative force or a conservative vector field if it meets one of three equivalent conditions.
– One condition is that the curl of the force field is the zero vector, indicating that there is no rotation or circulation of the force.
– Another condition is that the net work done by the force when moving a particle through a closed trajectory is zero.
– The third condition is that the force can be expressed as the negative gradient of a potential.
– These three conditions are mathematically equivalent and establish the conservative nature of a force field.
– However, not all forces satisfy these conditions, such as velocity-dependent forces like friction, which are unambiguously non-conservative.

Non-conservative Forces
– Despite the conservation of total energy, non-conservative forces can arise in classical physics due to neglected degrees of freedom or time-dependent potentials.
– Non-conservative forces can be perceived as macroscopic effects of small-scale conservative forces.
– Friction, for example, can be treated without violating conservation of energy by considering the motion of individual molecules.
– However, non-conservative forces require considering all the individual forces at a microscopic level, making it more complex.
– While some forces, like the magnetic force, may satisfy one or two conditions of a conservative force, they are often classified as non-conservative due to not meeting all three conditions.

Conservative Vector Field
– A conservative vector field is a vector field whose curl is zero.
– In a conservative vector field, the line integral along any closed path is zero.
– Examples of conservative vector fields include gravitational and electrostatic fields.
– Conservative vector fields have potential functions that can be used to calculate work and energy.
– Conservative vector fields satisfy the fundamental theorem of calculus for line integrals.

Conservative System
– A conservative system is a physical system whose total mechanical energy is conserved.
– In a conservative system, the work done by non-conservative forces is zero.
– Examples of conservative systems include a pendulum and a mass-spring system.
– Conservation of mechanical energy in a conservative system is a consequence of the principle of conservation of energy.
– Conservative systems can be described using the concept of potential energy.

Examples and Additional Resources
– Gravitational force is a conservative force that depends on the masses and distances between objects.
– Electrostatic force is a conservative force that depends on the charges and distances between objects.
– Elastic force in a spring is a conservative force that depends on the displacement of the spring.
– Magnetic force can be considered a conservative force as it acts perpendicular to velocity, resulting in zero work done.
– Conservative forces can be found in various physical systems, such as pendulums and harmonic oscillators.
– Additional resources for further study and exploration of conservative forces include books like ‘Analytical Mechanics’ by Louis N. Hand and Janet D. Finch, ‘Classical Mechanics’ by John R. Taylor, ‘Mechanics’ by P. K. Srivastava, ‘The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory’ by Rüdiger and Hollerbach, ‘Klassische Mechanik’ by Friedhelm Kuypers, and ‘Classical Mechanics’ by Tom W. B. Kibble and Frank H. Berkshire. The Wikipedia page on conservative forces also provides further information and references on the topic. Source:  https://en.wikipedia.org/wiki/Conservative_force