Why Gravitational Mass is Related to Inertial Mass
San José State University

Thayer Watkins
Silicon Valley
& Tornado Alley

Why Gravitational Mass
is Related to Inertial Mass


Isaac Newton formulated dynamics in terms of the equation force is equal to mass times acceleration, or in symbols

F = ma

This means the inertial mass of a physical body is its resistance to acceleration.

Newton also formulated the Law of Gravitation: The force between two bodies is proportional to the product of their masses divided by the distance between them squared.

There is no logical requirement that the m for a body in F=ma should be the same as the m for it in the Law of Gravitation. But it is and this material explores why this amazing aspect of the physical world should be true.

An Implication

Let M and m be the gravitational masses of two astronomical bodies. The force acting on each due to the gravitational attraction between them is

F = GMm/s²

where G is the gravitaional constant and s is the separation distance between the centers of the two bodies.

Suppose that M is so large compared to m that the smaller body travels in a circular orbit of radius s around the larger one. This makes s equal to r.

The centripetal acceleration a for such a circular orbit is

a = v²/r = ω²r

where v is tangential velocity and ω is angular velocity in radians per second. Suppose mi is inertial mass. Then the gravitational attraction between the two bodies must match mia for the smaller body; i.e.,

GMm/s² = miv²/r = miω²r

If the inertial mass is the same as the gravitational mass

v²/r = ω²r = GM/r²
and hence
v² = GM/r
ω² = GM/r³

This relation means that any body in an orbit of radius r travels at the same speed. Since the orbit period T is given by 2πr/v

T = 2πr3/2/(GM)½

any body in an orbit of radius r has the same period T.

The stability of the solar system is proof that gravitational mass is equal to inertial mass.

Another Implication

Again let M and m be the gravitational masses of two bodies. The force acting on the body of mass m is

F = GMm/s²

Thus the acceleration experienced by that body is given by

ma = GMm/s²
and hence
a = GM/s²

This is independent of m. All bodies being gravitationally attracted by the body of mass M are subjected to the same accleration. A group of bodies located at the same distance from M and subjected to its attraction would move as a unit and get to a specified distance at the same time, as Galileo Galilei demonstrated for the Earth in about 1686.

It would be a quite different world if gravitational mass were not related to inertial mass.

The Equivalence Principle

Albert Einstein articulated the principle that it is impossible to to distinguish between bodies being subjected to gravitational attraction and bodies being subjected to uniformly constant acceleration. The question is why this should be so.

Gravitational Fields and Their Movement

A spherical mass of m establishes a spherically symmetrical field of intensity g that is radially oriented. Its magnitude at a point a distance r from the center of the mass is given by

g = Gm/r

where G is the gravitational constant.

If the mass m is moving its field has to be moving also. The process of the field moving involves perturbations that are not propogated instantaneously. Those perturbations can be interpreted as particles.

If the mass is moving at a contant velocity it seems that its field moves effortlessly. But it is known from the Special Theory of Relativity that the apparent mass increases; i.e.,

m = m0/(1 − (v/c)²)½

In words the inertial mass of the moving body increases with velocity.

In the literature on dynamics there are what are called the longitudinal and the transverse masses for a body; i.e.,

ml = m0/(1−β²)3/2
mt = m0/(1−β²)1/2

where longitudinal means in the direction of the motion and tranverse is perpendicular to the direction of motion. Transverse mass is just the relativistic mass and longitudinal mass is just the relativistic mass divided by (1−β²).

Surprisingly the concepts and terms longitudinal mass and transverve mass preceded Special Relativity. In the late 19th century various theorists noticed that charged bodies resisted acceleration more than is acounted for by their masses. J.J. Thompson pointed this out in 1881 and Oliver Heavyside worked out the mathematics of it in 1897.

But if the mass is moving at a contant acceleration the situation may be different.

The question of the effect of accelerated motion of fields may be examined by looking at accelerated movement of electromagnetic fields.

The dynamics of electromagnetic fields are given by Maxwell's Equations.

(To be continued.)

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