What is a Central Force?

A central force is a force with a direction and magnitude that only depend on an object's center and the distance from its center to another point. The direction a central force acts must be along the line joining the object's center to the other point, while the force's magnitude only depends on the distance, or radius, between the two. Examples of central forces are found in the gravitational force, the electrostatic force, and the force from a spring. Only two types of central forces can result in orbital motion: gravitational force and analogous force.

Newton’s law of universal gravitation states that the gravitational force between two objects is always directed toward one another. Furthermore, the magnitude of the force is inversely proportional to the square of the distance between the objects—in other words, doubling the distance between the objects will result in a force only one fourth as strong. When one object is much more massive than the other, the arrangement meets the criteria of central force.

An analogous central force is the electrostatic force between particles with an electric charge. Like the gravitational force, the electrostatic force is inversely proportional to the square of the distance between two particles. Unlike gravity, however, electrostatic interactions are proportional to the product of their charges, rather than their masses. They also tend to dominate on very small scales. Mathematically speaking, though, the magnitude of the gravitational force and the electrostatic force both follow the inverse-square law.

A different kind of central force can result from the interaction of a spring with an object. The force produced by a spring is proportional to the distance the spring is stretched from its equilibrium length. Stretching a spring twice the distance from its equilibrium length will result in an attractive force twice as strong. This kind of behavior is known as Hooke’s law and is seen not only in springs, but in any linearly elastic material—including most solid metals. When a spring or suitable material is anchored at a center, it too can meet the criteria of a central force.

Though increasing the distance between objects or particles has a very different effect in systems governed by the inverse-square law versus Hooke’s law, both of these central forces will produce closed orbital motion. Planets orbit the Sun because of gravity, a central force subject to the inverse-square law. Similarly, charged particles can orbit an oppositely-charged center due to the electrostatic force. A somewhat lesser-known fact is that a centrally-anchored spring can, when other forces are negligible, cause an object to orbit the center. Simple harmonic motion is one such example that is constrained to move along only one dimension.