# "Proving Central Force is a Conservative Force

• m2287
In summary, a central force is a force that is always directed towards a given point and its magnitude depends only on the distance from that point. Any central force is also a conservative force, meaning that no energy is lost when an object is moved around and placed back in its original position. To prove this, we use the general definition of work done by a force, which involves calculus, as the force is not constant due to its dependence on displacement.
m2287

## Homework Statement

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.

## The Attempt at a Solution

i really don't have a clue...

im guessing work done = force * distance

and F = ma might be used but i don't know how!

What is the definition of a conservative force?

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

m2287 said:
that if you move an object around and place it back in its original position no energy is lost?
Correct. Can you therefore write an expression for the work done by a central force?

wd= ma * d ?

m2287 said:
wd= ma * d ?
Close but don't forget that the magnitude of the force is dependant on the displacement. Use vector notation for the displacement and write ma as F.

does than not just give you wd = f* d

im very confused about how to prove this!

m2287 said:
does than not just give you wd = f* d
No. Note that the magnitude of the force is a function of displacement and note that we are looking at displacement which is a vector quantity. Now, because the force is not constant (because is varies with displacement), we cannot use the simple relationship you outlined above. Do you know a general definition of the work done by a force (involving calculus)?

## What is a central force?

A central force is a type of force that acts on a particle towards or away from a fixed point, known as the center of force. This type of force always acts along the line joining the particle and the center of force.

## What is a conservative force?

A conservative force is a type of force that does not depend on the path taken by the particle and only depends on the initial and final positions of the particle. This means that the work done by a conservative force on a particle is independent of the path taken by the particle.

## How can a central force be proven to be a conservative force?

A central force can be proven to be a conservative force by verifying that the work done by the force on a particle moving along any closed path is zero. If the work done is zero, then it indicates that the force is conservative.

## What is the significance of proving that a central force is a conservative force?

Proving that a central force is a conservative force allows us to use the principle of conservation of energy to analyze the motion of particles under the influence of such forces. This simplifies calculations and makes it easier to predict the behavior of these systems.

## Are all central forces conservative forces?

Yes, all central forces are conservative forces. This is because they satisfy the criteria for a conservative force, which is that the work done by the force is path independent. Therefore, any force that acts along the line joining the particle and the center of force is considered to be a conservative force.

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