Show that any central force is a conservative force

[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

 

[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.

 

[ch00se]

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.

thanks for the reply

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

 

[neutrino]

what is the more general definition of work?

$$\int_{A}^{B}\vec{F}.d\vec{s}$$

In this problem, both the limits of integration are the same, since the object ends up the point where it started from.

http://hyperphysics.phy-astr.gsu.edu/hbase/wint.html#wg

 

[ch00se]

$$\int_{A}^{B}\vec{F}.d\vec{s}$$

In this problem, both the limits of integration are the same, since the object ends up the point where it started from.

http://hyperphysics.phy-astr.gsu.edu/hbase/wint.html#wg

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?

 

[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?

 

[ch00se]

the work done is 0

and the integral = 0 ?

 

[neutrino]

the work done is 0

and the integral = 0 ?

Exactly. :)

 

[ch00se]

Exactly. :)

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?

 

[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.

 

[robgazza]

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.

they need it in cartesian lol

could you display that for me please?

appreciate your help!

 

[tim_lou]

simply find a potential function for the force field. the gradient field of any function is conservative.