# 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^{[1]}

- $ 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} $.^{[2]}

### 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

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

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