Is a Central Force Always Conservative?

AI Thread Summary
A central force, which acts along the line connecting two objects and depends solely on the distance between them, is classified as a conservative force. To demonstrate this in Cartesian coordinates, one must show that the work done by the force over a closed path is zero or that the work depends only on the initial and final positions. The force can be expressed as F = F(r)r, and the work can be calculated using the integral W = ∫F·dr. The discussion highlights the challenge of breaking the force into its x, y, and z components for the solution. Ultimately, understanding the properties of central forces is crucial for proving their conservative nature.
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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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