If a force is a central force and can be written as $\vec{F}(\vec{r})=f{(r)}\hat{r}$ , then it is a conservative force. But is the converse true? I mean, are all conservative forces a central force? If no, can you please provide explanation?
Answer
Good question.
First, to be clear on definitions, a conservative field of force is one where the work done between any two fixed points is independent of the path taken; and this is equivalent (at least in Euclidean space) to saying that the work done in any closed loop is zero.
Further, the sum of any two conservative fields is also conservative.
Now take the Earth-Moon system, then we can see quite directly that the gravitational force felt by some satellite being the sum of two conservative fields is also conservative, but can't be central to some fixed point: close to the moon, it's directed towards its centre and close to the earth it's directed to its.
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