Is a Central Force Always Conservative?

[subwaybusker]

Homework Statement

Show that a central force between two objects, ie. one that acts along the vector connecting their centres, r[tex]^{$$\Downarrow$$}[/tex],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 = $$\int$$F. 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.

 

[subwaybusker]

nobody?

 

[thomate1]

I would approach this problem by first defining what a conservative force is. A conservative force is one that does not depend on the path taken by an object, but only on the initial and final positions of the object. In other words, the work done by a conservative force is independent of the path taken by the object and only depends on the endpoints.

Now, let's consider a central force between two objects, where the force acts along the vector connecting their centers, r. Since the force is central, it can be expressed as F(r)r, where F(r) is the strength of the force and r is the unit vector in the direction of the force.

To show that this central force is conservative, we need to show that the work done by the force is independent of the path taken by the object and only depends on the endpoints. Let's consider two points, A and B, with position vectors rA and rB, respectively. The work done by the central force from A to B can be expressed as:

W = ∫F(r)r.dr from A to B

Since the force is central, we can express r in terms of the x, y, and z components as r = xi + yj + zk. Therefore, we can rewrite the above equation as:

W = ∫F(r)(xi + yj + zk).d(xi + yj + zk) from A to B

Expanding the dot product and using the chain rule, we get:

W = ∫(xFx + yFy + zFz)dx + (xFy + yFx)dy + (xFz + zFx)dz from A to B

Since the force only depends on r, we can rewrite Fx, Fy, and Fz as Fx(r), Fy(r), and Fz(r), respectively. Therefore, the above equation becomes:

W = ∫(xFx(r) + yFy(r) + zFz(r))dx + (xFy(r) + yFx(r))dy + (xFz(r) + zFx(r))dz from A to B

Now, since we are integrating from A to B, we can express dx, dy, and dz in terms of dr as dx = x'dr, dy = y'dr, and dz = z'dr, where x', y', and z' are the components of the unit vector