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 individual masses, it is times the harmonic mean of these masses. ${\displaystyle m_{\mathrm {red} }}$ ${\displaystyle N}$ ${\displaystyle {\frac {1}{N}}}$

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 ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$

${\displaystyle {\frac {1}{m_{\mathrm {red} }}}={\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}}$

characterized reduced mass

${\displaystyle m_{\mathrm {red} }:={\frac {m_{1}\,m_{2}}{m_{1}+m_{2}}}}$

Has.

Depending on the mass of the heavier body ( ), the following applies to the reduced mass: ${\displaystyle m_{1}}$ ${\displaystyle m_{1}\geq m_{2}}$

${\displaystyle {\frac {m_{2}}{2}}\leq m_{\mathrm {red} }<m_{2}}$

with the marginal values

• ${\displaystyle m_{\mathrm {red} }\approx m_{2}/2}$for and ${\displaystyle m_{1}\approx m_{2}}$
• ${\displaystyle m_{\mathrm {red} }\approx m_{2}}$ for . ${\displaystyle m_{1}\gg m_{2}\Leftrightarrow m_{2}/m_{1}\ll 1}$

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: ${\displaystyle m_{2}/m_{1}\ll 1}$

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

For example, the relative motion 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 . ${\displaystyle m_{\mathrm {red} }}$

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

Derivation

• With vanishing total force, the equations of motion for the locations and the two bodies are: ${\displaystyle {\vec {r}}_{1}}$ ${\displaystyle {\vec {r}}_{2}}$

${\displaystyle m_{1}{\frac {\mathrm {d} ^{2}{\vec {r}}_{1}}{\mathrm {d} t^{2}}}={\vec {F}}}$

${\displaystyle m_{2}{\frac {\mathrm {d} ^{2}{\vec {r}}_{2}}{\mathrm {d} t^{2}}}=-{\vec {F}}}$

• Adding these two equations gives the center of gravity

${\displaystyle {\vec {R}}:={\frac {m_{1}{\vec {r}}_{1}+m_{2}{\vec {r}}_{2}}{M}}}$

with the mass sum the equation of motion ${\displaystyle M:=m_{1}+m_{2}}$

${\displaystyle {\ddot {\vec {R}}}=0}$

of a free particle. So the center of gravity moves in a straight line uniformly :

${\displaystyle {\vec {R}}(t)={\vec {R}}(0)+t\,{\vec {v}}(0)}$

• If one subtracts the equations of motion of the particles divided by the respective mass, one obtains

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}({\vec {r}}_{1}-{\vec {r}}_{2})=\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right){\vec {F}}={\frac {1}{m_{\mathrm {red} }}}{\vec {F}}}$

${\displaystyle \Leftrightarrow m_{\mathrm {red} }{\frac {\mathrm {d} ^{2}{\vec {r}}}{\mathrm {d} t^{2}}}={\vec {F}}}$

as the equation of motion for the relative position vector . This moves like a particle of reduced mass under the influence of the force . ${\displaystyle {\vec {r}}:={\vec {r}}_{1}-{\vec {r}}_{2}}$ ${\displaystyle m_{\mathrm {red} }}$ ${\displaystyle {\vec {F}}}$

Angular momentum

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

${\displaystyle {\begin{aligned}{\vec {L}}_{\mathrm {S} }&=\sum _{i=1}^{2}{\vec {L}}_{i\mathrm {S} }=({\vec {r}}_{1\mathrm {S} }\times {\vec {p}}_{1\mathrm {S} })+({\vec {r}}_{2\mathrm {S} }\times {\vec {p}}_{2\mathrm {S} })\\&=({\vec {r}}_{1\mathrm {S} }-{\vec {r}}_{2\mathrm {S} })\times {\vec {p}}_{1\mathrm {S} }={\vec {r}}_{12}\times m_{\mathrm {red} }{\vec {v}}_{1\mathrm {2} }\end{aligned}}}$

Designate here

• ${\displaystyle {\vec {r}}_{i\mathrm {S} },{\vec {p}}_{i\mathrm {S} }}$the position vector or the momentum of the particle in relation to the center of gravity. ${\displaystyle i}$
• ${\displaystyle {\vec {r}}_{12},{\vec {v}}_{12}}$ 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 . ${\displaystyle m_{\mathrm {red} }{\vec {v}}_{12}}$ ${\displaystyle {\vec {r}}_{12}}$

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 ${\displaystyle m}$ ${\displaystyle r_{\mathrm {m} }}$ ${\displaystyle r}$ ${\displaystyle r}$

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

the reduced mass is calculated as follows:

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

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