# 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}} 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 .