Reduced mass

In physics, reduced mass is a measure of the effective inertial mass of a system with two or more particles when the particles are interacting with each other.

Note, however, that the mass determining the gravitational force is not reduced.

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.

Reduced mass is particularly useful in classical mechanics.

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[1][2]

and has the reciprocal additive property:

which by re-arrangement is equivalent to half of the harmonic mean.

Using Newton's second law, the force exerted by a body (particle 2) on another body (particle 1) is:

The relative acceleration arel between the two bodies is given by:

Note that (since the derivative is a linear operator) the relative acceleration

This simplifies the description of the system to one force (since

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,

The potential energy V is a function as it is only dependent on the absolute distance between the particles.

and let the centre of mass coincide with our origin in this reference frame, i.e.

Thus we have reduced the two-body problem to that of one body.

Reduced mass can be used in a multitude of two-body problems, where classical mechanics is applicable.

This holds for a rotation around the center of mass.

The moment of inertia around this axis can be then simplified to

In a collision with a coefficient of restitution e, the change in kinetic energy can be written as

where vrel is the relative velocity of the bodies before collision.

For typical applications in nuclear physics, where one particle's mass is much larger than the other the reduced mass can be approximated as the smaller mass of the system.

In the case of the gravitational potential energy

we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass (M) equal to the one particular sum equal to the sum of these two masses , because

but all those other pairs whose sum is M would have the wrong product of their masses.

[3] They orbit each other about a common centre of mass, a two body problem.

To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass

This idea is used to set up the Schrödinger equation for the hydrogen atom.