# What is reduced mass in physics

## Reduced mass

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 \$ N \$ individual masses it is \$ \ frac {1} {N} \$ times the harmonic mean of these masses.

### Astronomy, particle motion

If two bodies with masses \$ m_1 \$ and \$ m_2 \$ move under the influence of a vanishing total force, the equations of motion can be split 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

\$ \ frac {1} {m_ \ mathrm {red}} = \ frac {1} {m_1} + \ frac {1} {m_2} \$

characterized reduced mass

\$ m_ \ mathrm {red}: = \ frac {m_1 m_2} {m_1 + m_2} \$

Has. Depending on the mass \$ m_1 \$ of the heavier body (\$ m_1 \ ge m_2 \$), the reduced mass \$ m_ \ mathrm {red} \$ has values ​​between \$ m_2 / 2 \$ and \$ m_2 \$. 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 (\$ m_2 / m_1 \ ll1 \$). Then the reduced mass is almost the mass of the lighter particle:

\$ m_ \ mathrm {red} = \ frac {m_2} {1 + m_2 / m_1} \ approx m_2 \ left (1- \ frac {m_2} {m_1} \ right) \ approx m_ \ mathrm 2 \$

For example, the relative motion moon-earth can be reduced to a one-body problem: The moon moves like a body with reduced mass \$ m_ \ mathrm {red} \$ in the earth's gravitational field.

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

### Derivation

• With vanishing total force, the equations of motion for the locations \$ \ vec {r} _1 \$ and \$ \ vec {r} _2 \$ of the two bodies are:
\$ m_1 \ frac {\ mathrm {d} ^ 2 \ vec {r} _1} {\ mathrm {d} t ^ 2} = \ vec {F} \$
\$ m_2 \ frac {\ mathrm {d} ^ 2 \ vec {r} _2} {\ mathrm {d} t ^ 2} = - \ vec {F} \$
• Added one obtains these two equations for the center of gravity
\$ \ vec {R}: = \ frac {m_1 \ vec {r} _1 + m_2 \ vec {r} _2} {M} \$
with the mass sum \$ M: = m_1 + m_2 \$ the equation of motion
\$ \ ddot {\ vec {R}} = 0 \$
of a free particle. So the center of gravity moves in a straight line uniformly:
\$ \ vec {R} (t) = \ vec {R} (0) + t \, \ vec {v} (0) \$
• Subtracted one obtains the equations of motion of the particles divided by the respective mass
\$ \ frac {\ mathrm {d} ^ 2} {\ mathrm {d} t ^ 2} (\ vec {r} _1- \ vec {r} _2) = \ left (\ frac {1} {m_1} + \ frac {1} {m_2} \ right) \ vec {F} = \ frac {1} {m_ \ mathrm {red}} \ vec {F} \$
\$ m_ \ mathrm {red} \ frac {\ mathrm {d} ^ 2 \ vec {r}} {\ mathrm {d} t ^ 2} = \ vec {F} \$
as the equation of motion for the relative position vector \$ \ vec {r}: = \ vec {r} _1- \ vec {r} _2 \$. This moves like a particle of reduced mass \$ m_ \ mathrm {red} \$ under the influence of the force \$ \ vec {F} \$.

### Angular momentum

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

\$ \ begin {align} \ vec L_ \ mathrm S & = \ sum_ {i = 1} ^ 2 \ vec L_ {i \ mathrm S} = (\ vec r_ {1 \ mathrm S} \ times \ vec p_ {1 \ mathrm S}) + (\ vec r_ {2 \ mathrm S} \ times \ vec p_ {2 \ mathrm S}) \ & = (\ vec r_ {1 \ mathrm S} - \ vec r_ {2 \ mathrm S}) \ times \ vec p_ {1 \ mathrm S} = \ vec r_ {12} \ times m_ \ mathrm {red} \ vec v_ {1 \ mathrm 2} \ end {align} \$
\$ \ vec r_ {i \ mathrm S}, \ vec p_ {i \ mathrm S} \$ denote the position vector or the momentum of the particle \$ i \$ related to the center of gravity.
\$ \ vec r_ {12}, \ vec v_ {12} \$ denote 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 large as the angular momentum of a particle with the momentum \$ m_ \ mathrm {red} \ vec v_ {12} \$ and the position vector \$ \ vec r_ {12} \$.

### Technical mechanics

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

\$ i = \ frac {r_ \ mathrm m} {r} \$

the reduced mass is calculated as follows:

\$ m_ \ mathrm {red} = i ^ 2 \, m \$

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 Google book search).
2. ↑ W.Demtröder: Experimental Physics 1. 7th edition. Springer-Verlag, Berlin 2015, ISBN 978-3-662-46415-1.