Centers of Gravity in Non-uniform Fields - Centers of Gravity in A Field

Centers of Gravity in A Field

When a body is affected by a non-uniform external gravitational field, one can sometimes define a center of gravity relative to that field that will act as a point where the gravitational force is applied. Textbooks such as the The Feynman Lectures on Physics characterize the center of gravity as a point about which there is no torque. In other words, the center of gravity is a point of application for the resultant force. Under this formulation, the center of gravity r_cg is defined as a point that satisfies the equation

where F and τ are the total force and torque on the body due to gravity.

One complication concerning r_cg is that its defining equation is not generally solvable. If F and τ are not orthogonal, then there is no solution; the force of gravity does not have a resultant and cannot be replaced by a single force at any point. There are some important special cases where F and τ are guaranteed to be orthogonal, such as if all forces lie in a single plane or are aligned with a single point.

If the equation is solvable, there is another complication: its solutions are not unique. Instead, there are infinitely many solutions; the set of all solutions is known as the line of action of the force. This line is parallel to the weight F. In general, there is no way to choose a particular point as the unique center of gravity. A single point may still be chosen in some special cases, such as if the gravitational field is parallel or spherically symmetric. These cases are considered below.