14.11: The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a point object is the vector sum of the gravitational forces on it due to various objects. The principle helps calculate the force by listing the individual forces and then vectorially summing them up. However, it should be noted that the principle of superposition is not always apparent. In the presence of a second force, the first force could have been amplified or reduced, or the two forces could have a combined effect that is a complex combination of the two. Thankfully, nature is simple in how forces affect a point object together.

The principle of superposition helps conceptualize the effect of gravitation in terms of a new concept, the gravitational field. In the vector addition of forces due to multiple masses, the mass of a test object is a scalar quantity that multiplies with the acceleration due to each force. It can be taken out of the summation and divided. The net force per unit mass then does not depend on the test mass. This quantity is called the gravitational field, a vector quantity whose direction is the same as the direction of the net gravitational force. By definition, the field follows the principle of superposition.

The gravitational field helps us understand the mathematical problem of studying the motion of objects under the influence of gravitational force. Various objects create a gravitational field around themselves; these add up vectorially to create a net gravitational field in space. When a test mass is present in the field, the net field creates a net force on it, which is the product of the field and its mass.

The gravitational field experienced by a particle of a certain mass due to another particle is the gravitational force acting on that particle divided by its mass.

The principle of superposition states that the net gravitational field at any point is the vector sum of the gravitational fields due to individual point objects.

A real object with a definite shape can be divided into infinitesimal parts, each having a differential mass producing a differential field at a certain distance. The resultant field due to the entire object is obtained according to the superposition principle by integrating over a suitable limit.

For example, consider a sphere with constant density. The differential mass for any constituent shell can be written as density times the differential volume.

The field equation for a shell is known, into which the differential mass is substituted.

The expression is then integrated from zero to the radius of the sphere.

Since density times volume gives the mass, the final field obtained is the same as that of a point mass.