# Show that any central force is a conservative force

1. Dec 7, 2006

### ch00se

hello, i am having problems with this question

"If a force on an object is always directed along a line from the object to a given point, and the magnitude of the force depends only on the distance of the object from the point, the force is said to be a central force. Show that any central force is a conservative force."

i know that if you move an object around and place it back in its original position no energy is lost

i have wd= ma * d

however i would need to write ma as F ?

very stuck, thanks

Last edited: Dec 7, 2006
2. Dec 7, 2006

### neutrino

It is best to do this problem in polar coordinates. In spherical coordinates, a central force has the form $$F(r)\hat{r}$$. Now, use the more general definition of work (i.e. as an integral) to prove that it is a conservative force.

3. Dec 7, 2006

### ch00se

thanks for the reply

im not sure i follow 100%, what is the more general definition of work?

4. Dec 7, 2006

5. Dec 7, 2006

### ch00se

ok, i understand, however how would i go about proving this?

would i need some numerical evidence or would equations without a definite answer suffice?

6. Dec 7, 2006

### neutrino

hint: There is no net change in energy. What does this say about the amount of work done in the closed path, and hence the integral?

7. Dec 7, 2006

### ch00se

the work done is 0

and the integral = 0 ?

8. Dec 7, 2006

Exactly. :)

9. Dec 7, 2006

### ch00se

thanks for your help, much appreciated

this is a coursework question worth a lot of marks, 20 in fact.

can you give me some pointers on what to include in my answer please?

10. Dec 7, 2006

### neutrino

Here's how I'd solve it, but if your course demands that you solve it in some other way (like using cartesian instead of polar, etc), this may not be useful.

Take the "given point" as the origin of your coordiante system.

For a force to be conservative, the work done by it on an object around any closed path should be zero.

$$W = \int_{A}^{A}\vec{F}.d\vec{s} = 0$$

As stated earlier, $$\vec{F} = F(r)\hat{r}$$, where $$\hat{r}$$ is the unit vector in the radial direction.

$$d\vec{s} = d\hat{r} + rd\theta\hat{\theta} + r\sin{\phi}d{\phi}\hat{\phi}$$

Now solve the integral.

11. Dec 7, 2006

### robgazza

they need it in cartesian lol

could you display that for me please?