Reduced Mass
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the reduced mass is the "effective"
inertial mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
appearing in the
two-body problem In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
of
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
. 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 In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
is ''not'' reduced. In the computation, one mass ''can'' be replaced with the reduced mass, if this is compensated by replacing the other mass with the sum of both masses. The reduced mass is frequently denoted by \mu ( mu), although the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when ...
is also denoted by \mu (as are a number of other physical quantities). It has the
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
of mass, and
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
kg.


Equation

Given two bodies, one with mass ''m''1 and the other with mass ''m''2, 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 :\mu = \cfrac = \cfrac,\!\, 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 = \frac + \frac \,\! which by re-arrangement is equivalent to half of the
harmonic mean In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the recipro ...
. In the special case that m_1 = m_2: : = \frac = \frac\,\! If m_1 \gg m_2, then \mu \approx m_2.


Derivation

The equation can be derived as follows.


Newtonian mechanics

Using
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
, the force exerted by a body (particle 2) on another body (particle 1) is: :\mathbf_ = m_1 \mathbf_1 The force exerted by particle 1 on particle 2 is: :\mathbf_ = m_2 \mathbf_2 According to
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
, the force that particle 2 exerts on particle 1 is equal and opposite to the force that particle 1 exerts on particle 2: :\mathbf_ = - \mathbf_ Therefore: :m_1 \mathbf_1 = - m_2 \mathbf_2 \;\; \Rightarrow \;\; \mathbf_2=- \mathbf_1 The relative acceleration arel between the two bodies is given by: :\mathbf_ := \mathbf_1-\mathbf_2 = \left(1+\frac\right) \mathbf_1 = \frac m_1 \mathbf_1 = \frac Note that (since the derivative is a linear operator), the relative acceleration \mathbf_ is equal to the acceleration of the separation \mathbf_ between the two particles. :\mathbf_ = \mathbf_1-\mathbf_2 = \frac - \frac = \frac(\mathbf_1 - \mathbf_2) = \frac This simplifies the description of the system to one force (since \mathbf_ = - \mathbf_), one coordinate \mathbf_, and one mass \mu. Thus we have reduced our problem to a single degree of freedom, and we can conclude that particle 1 moves with respect to the position of particle 2 as a single particle of mass equal to the reduced mass, \mu.


Lagrangian mechanics

Alternatively, a Lagrangian description of the two-body problem gives a
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of : \mathcal = m_1 \mathbf_1^2 + m_2 \mathbf_2^2 - V(, \mathbf_1 - \mathbf_2 , ) \!\, where _ is the position vector of mass m_ (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 = \mathbf_1 - \mathbf_2 and let the centre of mass coincide with our origin in this reference frame, i.e. : m_1 \mathbf_1 + m_2 \mathbf_2 = 0 , then : \mathbf_1 = \frac , \; \mathbf_2 = -\frac. Then substituting above gives a new Lagrangian : \mathcal = \mu \mathbf^2 - V(r), where :\mu = \frac 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.


Moment of inertia of two point masses in a line

In a system with two point masses m_1 and m_2 such that they are co-linear, the two distances r_1 and r_2 to the rotation axis may be found with r_1 = R \frac r_2 = R \frac where R is the sum of both distances R = r_1 + r_2 . This holds for a rotation around the center of mass. The
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
around this axis can be then simplified to I = m_1 r_1^2 + m_2 r_2^2 = R^2 \frac + R^2 \frac = \mu R^2.


Collisions of particles

In a collision with a
coefficient of restitution The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectl ...
''e'', the change in kinetic energy can be written as :\Delta K = \frac\mu v^2_(e^2-1), where vrel is the relative velocity of the bodies before
collision In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great fo ...
. 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 particle's exact mass is not known.


Motion of two massive bodies under their gravitational attraction

In the case of the gravitational potential energy :V(, \mathbf_1 - \mathbf_2 , ) = - \frac \, , 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 The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
(mass ''me'') and
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
(mass ''mp'') in the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
. 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 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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for the hydrogen atom.


Other uses

"Reduced mass" may also refer more generally to an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ic term of the form :x^* = = \!\, that simplifies an equation of the form :\ = \sum_^n = + + \cdots+ .\!\, The reduced mass is typically used as a relationship between two system elements in parallel, such as
resistors A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
; whether these be in the electrical, thermal, hydraulic, or mechanical domains. A similar expression appears in the transversal vibrations of beams for the elastic moduli.Experimental study of the Timoshenko beam theory predictions, A.Díaz-de-Anda J.Flores, L.Gutiérrez, R.A.Méndez-Sánchez, G.Monsivais, and A.Morales.Journal of Sound and Vibration Volume 331, Issue 26, 17 December 2012, Pages 5732-5744 https://doi.org/10.1016/j.jsv.2012.07.041 This relationship is determined by the physical properties of the elements as well as the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
linking them.


See also

*
Center-of-momentum frame In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...
*
Momentum conservation In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
*
Defining equation (physics) In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units. Description of units and physical quantities Physical q ...
*
Harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
*
Chirp mass In astrophysics the chirp mass of a compact binary system determines the leading-order orbital evolution of the system as a result of energy loss from emitting gravitational waves Gravitational waves are waves of the intensity of gravity genera ...
, a relativistic equivalent used in the
post-Newtonian expansion In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...


References

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External links


''Reduced Mass'' on HyperPhysics
Mechanics Mass