# What do we understand by a force being central?

Gravity is said to be a central force. But the resultant force field of multiple bodies is no longer central as it has many attraction points. My Doubts:

• Is the idea of a force being central limited only to two bodies?
• Is it possible to have a central force field with more than two particles?
• Does a force cease to be central when the number of particles in the system is increased?
• Is centripetal force Central?

note: my understanding of physics is under-graduate level.

• A central force is a force that only depends on the distance between the two objects and nothing else. Gravity is central because the law of gravitation states that the force is proportional to 1/r with no other spatial dependence. Another analogue would be the Coulomb law. The gravitational force field of multiple bodies is not central because it depends on the distance from multiple objects, and this can not be represented as a function of only one distance. – Ezze Oct 15 '19 at 13:52
• @Ezze Please post answers as answers, not comments – Aaron Stevens Oct 15 '19 at 14:50

## 2 Answers

A central force is simply a force that is always directed towards a fixed point in space. Gravity can be treated as a central force in certain circumstances.

The gravitational force acting on object $$A$$ orbiting around object $$B$$ can be approximated by a central force acting towards the centre of $$B$$ if $$B$$ is much more massive than $$A$$. By Newton's third law the gravitational attraction of $$A$$ on $$B$$ is equal and opposite to that of $$B$$ on $$A$$, but if $$B$$ is much more massive than $$A$$ then to a first approximation we can assume that $$B$$ is stationary.

It is also true that in the two-body problem, where $$A$$ and $$B$$ have similar masses, then they will orbit around a fixed point or barycenter. The gravitational attraction of one object on the other still acts as a central force directed towards this barycenter, but the barycenter is no longer at the centre of one of the objects. The existence of a stationary barycenter for the two-body problem is not obvious, and must be proved.

In the more general n-body problem there is no fixed point towards which the net gravitational attraction on each body always acts. So gravity can no longer be treated as a central force - this makes the n-body problem much more difficult to solve.

A central force is just one that acts in the direction of a particular central point. I think you need to distinguish in your mind the difference between a field or potential and a force. A massive body will exert a central force of gravity on another massive body. If you have several bodies together they will each continue to exert central forces individually, but their combined effect will not generally be directed to any meaningful centre. Moreover, the direction of their cumulative effect will vary from place to place, unlike the gravitational force due to a single body, which is always directed at its centre. However, it is possible for multiple bodies to produce a central force at certain points in space, depending on their spatial distribution.

A centripetal force is by definition one that is always directed at a central point.