Is a Central Force Always Conservative?

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SUMMARY

A central force, defined as a force that acts along the vector connecting the centers of two objects and depends solely on the distance r, is inherently conservative. This conclusion is derived from the work done by the force, which can be expressed as W = ∫F·dr. In Cartesian coordinates, the force can be decomposed into its x, y, and z components, confirming that the work done is path-independent and depends only on the initial and final positions.

PREREQUISITES
  • Understanding of vector calculus and force vectors
  • Familiarity with conservative forces and potential energy concepts
  • Knowledge of Cartesian coordinate systems
  • Proficiency in evaluating line integrals
NEXT STEPS
  • Study the properties of conservative forces in classical mechanics
  • Learn how to compute line integrals in Cartesian coordinates
  • Explore the relationship between work and potential energy
  • Investigate examples of central forces, such as gravitational and electrostatic forces
USEFUL FOR

Students of physics, particularly those studying classical mechanics, as well as educators and anyone interested in the mathematical foundations of force and energy concepts.

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Homework Statement


Show that a central force between two objects, ie. one that acts along the vector connecting their centres, r^{\Downarrow},with a strength that depends on only r, is a conservative force. I am supposed to do this in cartesian coordinates and show that the work is zero or that the work only depends on the endpoints.


Homework Equations





The Attempt at a Solution



I know that F=F(r)r, and that W = \intF. dr from A to A = 0. I tried breaking it into x,y,z components because of the cartesian hint, but I don't know what to do from there.
 
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