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Concept
Reduced mass
Summary
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass determining the gravitational force is not reduced. In the computation, one mass can be replaced with the reduced mass, if this is compensated by replacing the other mass with the sum of both masses. The reduced mass is frequently denoted by (mu), although the standard gravitational parameter is also denoted by (as are a number of other physical quantities). It has the dimensions of mass, and SI unit kg.
Given two bodies, one with mass m1 and the other with mass m2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass
where the force on this mass is given by the force between the two bodies.
The reduced mass is always less than or equal to the mass of each body:
and has the reciprocal additive property:
which by re-arrangement is equivalent to half of the harmonic mean.
In the special case that :
If , then .
The equation can be derived as follows.
Newtonian mechanics
Using Newton's second law, the force exerted by a body (particle 2) on another body (particle 1) is:
The force exerted by particle 1 on particle 2 is:
According to Newton's third law, the force that particle 2 exerts on particle 1 is equal and opposite to the force that particle 1 exerts on particle 2:
Therefore:
The relative acceleration arel between the two bodies is given by:
Note that (since the derivative is a linear operator) the relative acceleration is equal to the acceleration of the separation between the two particles.
This simplifies the description of the system to one force (since ), one coordinate , and one mass . Thus we have reduced our problem to a single degree of freedom, and we can conclude that particle 1 moves with respect to the position of particle 2 as a single particle of mass equal to the reduced mass, .
Official source
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