Reduced mass

In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass determining the gravitational force is not reduced. In the computation one mass can be replaced by the reduced mass, if this is compensated by replacing the other mass by the sum of both masses. The reduced mass is frequently denoted by $\mu$ (Greek lower case mu), although the standard gravitational parameter is also denoted by $\mu$ (and so are a number of other physical quantities as well). It has the dimensions of mass, and SI unit kg.

Equation

Given two bodies, one with mass m₁ and the other with mass m₂, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass ^[1]^[2]

\mu = \cfrac{1}{\cfrac{1}{m_1}+\cfrac{1}{m_2}} = \cfrac{m_1 m_2}{m_1 + m_2},\!\,

where the force on this mass is given by the force between the two bodies.

Properties

The reduced mass is always less than or equal to the mass of each body:

\mu \leq m_1, \quad \mu \leq m_2 \!\,

and has the reciprocal additive property:

\frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2} \,\!

which by re-arrangement is equivalent to half of the harmonic mean.

In the special case that $m_1 = m_2$ :

{\mu} = \frac{m_1}{2} = \frac{m_2}{2}\,\!

Derivation

The equation can be derived as follows.

Newtonian mechanics

Main article: Newtonian mechanics

Using Newton's second law, the force exerted by body 2 on body 1 is

\bold{F}_{12} = m_1 \bold{a}_1. \!\,

The force exerted by body 1 on body 2 is

\bold{F}_{21} = m_2 \bold{a}_2. \!\,

According to Newton's third law, the force that body 2 exerts on body 1 is equal and opposite to the force that body 1 exerts on body 2:

\bold{F}_{12} = - \bold{F}_{21}.\!\,

Therefore,

m_1 \bold{a}_1 = - m_2 \bold{a}_2. \!\,

and

\bold{a}_2=-{m_1 \over m_2} \bold{a}_1. \!\,

The relative acceleration a_rel between the two bodies is given by

\bold{a}_{\rm rel}= \bold{a}_1-\bold{a}_2 = \left(1+\frac{m_1}{m_2}\right) \bold{a}_1 = \frac{m_2+m_1}{m_1 m_2} m_1 \bold{a}_1 = \frac{\bold{F}_{12}}{m_{\rm red}}.

So we conclude that body 1 moves with respect to the position of body 2 as a body of mass equal to the reduced mass.

Lagrangian mechanics

Main article: Lagrangian mechanics

Alternatively, a Lagrangian description of the two-body problem gives a Lagrangian of

L = {1 \over 2} m_1 \mathbf{\dot{r}}_1^2 + {1 \over 2} m_2 \mathbf{\dot{r}}_2^2 - V(| \mathbf{r}_1 - \mathbf{r}_2 | ) \!\,

where ${\mathbf{r}}_{i}$ is the position vector of mass $m_{i}$ (of particle $i$ ). The potential energy V is a function as it is only dependent on the absolute distance between the particles. If we define

\mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2

and let the centre of mass coincide with our origin in this reference frame, i.e.

m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2 = 0

then

\mathbf{r}_1 = \frac{m_2 \mathbf{r}}{m_1 + m_2} , \mathbf{r}_2 = \frac{-m_1 \mathbf{r}}{m_1 + m_2}.

Then substituting above gives a new Lagrangian

L = {1 \over 2}\mu \mathbf{\dot{r}}^2 - V(r),

where

\mu = \frac{m_1 m_2}{m_1 + m_2}

is the reduced mass. Thus we have reduced the two-body problem to that of one body.

Applications

Reduced mass can be used in a multitude of two-body problems, where classical mechanics is applicable.

Collisions of particles

In a collision with a coefficient of restitution e, the change in kinetic energy can be written as

\Delta K = \frac{1}{2}\mu v^2_{\rm rel}(e^2-1)

where v_rel is the relative velocity of the bodies before collision.

For typical applications in nuclear physics, where one particle's mass is much larger than the other the reduced mass can be approximated as the smaller mass of the system. The limit of the reduced mass formula as one mass goes to infinity is the smaller mass, thus this approximation is used to ease calculations, especially when the larger particles exact mass is not known.

Motions of masses in gravitational fields

In the case of the gravitational potential energy

V(| \mathbf{r}_1 - \mathbf{r}_2 | ) = - \frac{G m_1 m_2}{| \mathbf{r}_1 - \mathbf{r}_2 |} \, ,

we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because

m_1 m_2 = (m_1+m_2) \mu\!\,

Non-relativistic quantum mechanics

Consider the electron (mass m_e) and proton (mass m_p) in the hydrogen atom.^[3] They orbit each other about a common centre of mass, a two body problem. To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass

m_e \rightarrow \frac{m_em_p}{m_e+m_p}

and the proton mass becomes the sum of the two masses

m_p \rightarrow m_e + m_p

This idea is used to set up the Schrödinger equation for the hydrogen atom.

Other uses

"Reduced mass" may also refer more generally to an algebraic term of the form

x^* = {1 \over {1 \over x_1} + {1 \over x_2}} = {x_1 x_2 \over x_1 + x_2}\!\,

that simplifies an equation of the form

\ {1\over x^*} = \sum_{i=1}^n {1\over x_i} = {1\over x_1} + {1\over x_2} + \cdots+ {1\over x_n}.\!\,

The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. This relationship is determined by the physical properties of the elements as well as the continuity equation linking them.

References

↑ Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3
↑ Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
↑ Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0

External links

Reduced Mass on HyperPhysics

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