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?
$\begingroup$
$\endgroup$
9
asked Oct 17, 2015 at 12:23
-
$\begingroup$ Gravity on flat earth surface. $\endgroup$– ariveroCommented Oct 17, 2015 at 12:30
-
3$\begingroup$ Any force that admits a potential is conservative. To get a non-central, conservative force, just pick a function $V(r, \theta, \phi)$, such that $\partial_\theta V \ne 0$ or $\partial_\phi V \ne 0$. Then $\vec F = -\nabla V$ will be a conservative, non-central force. One simple example $V=\frac 1 2 k x^2$ leads to the conservative force $\vec F = -\vec e_x kx$ which is obviously not central. $\endgroup$– Sebastian RieseCommented Oct 17, 2015 at 12:38
-
1$\begingroup$ @SebastianRiese These links say that $\vec F= -kx$ is a central conservative force..... books.google.co.in/… and sfu.ca/~boal/211lecs/211lec14.pdf $\endgroup$– SchrodingersCatCommented Oct 17, 2015 at 14:31
-
$\begingroup$ @Aniket Just clicked the sfu.ca link, it talk abouk $\vec F = -k\vec x$, that is a different beast (and I would have written it $\vec F = -k \vec r$ ... using $\vec x$ for positions vectors just generates confusion in my opinion $\vec r$ is avoids this). $\endgroup$– Sebastian RieseCommented Oct 17, 2015 at 14:32
-
$\begingroup$ @SebastianRiese What kind of a different beast? $\endgroup$– SchrodingersCatCommented Oct 17, 2015 at 14:33
|
Show 4 more comments
1 Answer
$\begingroup$
$\endgroup$
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.
answered Oct 18, 2015 at 10:28