Reduced mass

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The reduced mass is a fictitious mass which, under certain conditions, represents the properties of two individual masses of a system. Generalized for a system with individual masses, it is times the harmonic mean of these masses.

Astronomy, particle motion

If two bodies move with masses and without being subject to the influence of a total force, the equations of motion can be split up into the free movement of the center of gravity and the one-body problem of relative movement. The lighter particle behaves in the relative distance to the heavier particle like a particle that passes through

characterized reduced mass


Depending on the mass of the heavier body ( ), the following applies to the reduced mass:

with the marginal values

  • for and
  • for .

In important cases ( planetary motion , motion of an electron in the Coulomb field of the atomic nucleus ) the masses of the heavier and the lighter body differ very strongly ( ). Then the reduced mass is almost the mass of the lighter particle:

For example, the relative motion of the moon-earth can be reduced to a one-body problem: the moon moves like a body with reduced mass in the earth's gravitational field .

In many textbooks, the reduced mass is abbreviated with the Greek letter .


  • With vanishing total force, the equations of motion for the locations and the two bodies are:
  • Adding these two equations gives the center of gravity
with the mass sum the equation of motion
of a free particle. So the center of gravity moves in a straight line uniformly :
  • If one subtracts the equations of motion of the particles divided by the respective mass, one obtains
as the equation of motion for the relative position vector . This moves like a particle of reduced mass under the influence of the force .

Angular momentum

For a system of two particles, the angular momentum in the center of gravity system can be given as

Designate here

  • the position vector or the momentum of the particle in relation to the center of gravity.
  • the relative distance or the relative speed of the two particles.

In relation to the center of gravity, the angular momentum of a total system of two particles is exactly as great as the angular momentum of a particle with the momentum and the position vector .

Technical mechanics

A point mass that rotates around an axis at a distance can be converted to another distance . The reduced mass at the new distance has the same moment of inertia with respect to the axis of rotation as the original mass. With the translation

the reduced mass is calculated as follows:

Application e.g. B. in vibration theory .

Individual evidence

  1. C. Czeslik, H. Seemann, R. Winter: Basic knowledge of physical chemistry . 4th edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0937-7 ( limited preview in the Google book search).
  2. W.Demtröder : Experimentalphysik 1 . 7th edition. Springer-Verlag, Berlin 2015, ISBN 978-3-662-46415-1 .