Central forces act only on the center of an object and the distance to another point. Gravitational and electrostatic forces are examples of central forces that can cause orbital motion. Hooke’s law can also produce a central force. Both inverse square law and Hooke’s law can produce closed orbital motion.
A central force is a force whose direction and magnitude depend only on the center of an object and the distance from its center to another point. The direction in which a central force acts must be along the line joining the object’s center to the other point, while the magnitude of the force depends only on the distance, or radius, between the two. Examples of central forces are found in gravitational force, electrostatic force, and the force of a spring. Only two kinds of central forces can cause orbital motion: the gravitational force and the analogous force.
Newton’s law of universal gravitation states that the gravitational force between two objects is always directed towards each other. Also, 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-quarter as strong. When one object is much more massive than the other, the arrangement meets the central force criteria.
An analogous central force is the electrostatic force between electrically charged particles. 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, however, the magnitude of the gravitational force and the electrostatic force both follow the inverse square law.
A different type of core force can arise 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 type of behavior is known as Hooke’s law and is observed not only in springs, but in any linearly elastic material, including most solid metals. When a spring or suitable material is anchored to a center, it too can meet the criteria for a central force.
Although increasing the distance between objects or particles has a very different effect in systems governed by the inverse square law than by Hooke’s law, both of these central forces will produce closed orbital motion. Planets orbit the Sun due to gravity, a central force subject to the inverse square law. Similarly, charged particles can orbit a center of opposite charge due to electrostatic force. A somewhat lesser known fact is that a spring anchored at the center can, when other forces are negligible, cause an object to orbit around the center. Simple harmonic motion is one such example that is forced to move along only one dimension.
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