# Show that any central force is a conservative force

• ch00se
In summary: F(x,y) = (x+y)\hat{x} + (x-y)\hat{y}, then the potential function would be U(x,y) = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2 + C, where C is a constant. then, to prove that the force is conservative, find the partial derivatives of the potential function and show that they match the components of the force equation (i.e. \vec{F} = \nabla U). hope this helps

#### 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

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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.

neutrino said:
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.

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

ch00se said:
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

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

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?

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?

the work done is 0

and the integral = 0 ?

ch00se said:
the work done is 0

and the integral = 0 ?

Exactly. :)

neutrino said:
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?

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.

neutrino said:
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?

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

## 1. What is a central force?

A central force is a type of force that acts on an object in a direction that is always directed towards a fixed point, known as the center of force. This means that the magnitude and direction of the force only depend on the distance between the object and the center of force, and not on the object's orientation or position in space.

## 2. What is a conservative force?

A conservative force is a type of force that does not dissipate energy as it acts on an object. This means that the total energy of the object, including both kinetic and potential energy, remains constant as the object moves under the influence of a conservative force.

## 3. How do you show that a central force is a conservative force?

In order to show that a central force is conservative, you must demonstrate that the work done by the force is path-independent. This means that the work done by the force is the same regardless of the path taken by the object. One way to prove this is by using the fundamental theorem of calculus and showing that the force can be expressed as the gradient of a potential function.

## 4. What is the significance of a force being conservative?

A conservative force is significant because it allows for the conservation of energy. This means that the total energy of a system remains constant, and energy is not lost due to the force acting on the object. Additionally, conservative forces have many practical applications, such as in the design of efficient machines and devices.

## 5. Are all central forces conservative?

Yes, all central forces are conservative. This is because central forces are always directed towards a fixed point, which allows for the expression of the force as the gradient of a potential function. Therefore, the work done by a central force is path-independent, making it a conservative force.