Reduced mass

From Wikipedia, the free encyclopedia

Reduced mass is the "effective" mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the units of mass, which allows the two-body problem to be solved as if it were a one-body problem.

Given two bodies, one with mass m1 and the other with mass m2, they will orbit the barycenter of the two bodies. 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

m_{red} = {1 \over {{1 \over m_1} + {1 \over m_2}}} = {{m_1 m_2} \over {m_1 + m_2}}\ ,

where the force on this mass is given by the gravitational force between the two bodies. The reduced mass is frequently denoted by the Greek letter μ.

Applying the gravitational formula we get that the position of the first body with respect to the second is governed by the same differential equation as the position of a very small body orbiting a body with a mass equal to the sum of the two masses, because

{{m_1 m_2} \over {m_{red}}}=m_1+m_2.

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


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

x_{red} = {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_{eq}} = \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.


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