How to Calculate Reduced Mass

Written by paul dohrman
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When two bodies revolve around each other, they revolve around the shared centre of mass between them. For example, the Earth and Moon revolve around a point between their centres. The Moon causes the Earth to wobble. This complicates equations, for example the solution of orbital period. This same problem comes up with regard to an electron orbiting around a nucleus. The solution to this problem, called the “reduced mass,” therefore applies to both the very large in nature as well as the very small. The solution is to find a system that has the same frequency solution but is simpler to calculate. That simpler solution is to pretend the larger body is stationary in the centre and the smaller body orbits with a “reduced mass” at the same distance from the larger object as in the unmodified problem. The two-body problem then reduces to a one-body problem, focused solely on the smaller body’s orbit.

Skill level:
Easy

Instructions

1. 1

Calculate the reciprocals of the two bodies’ masses, using the same unit of mass for both.

For example, define the mass of an electron as 1 unit. A proton therefore has a mass of 1,836 units. The reciprocals are then 1/1 and 1/1836.

2. 2

The example above gives 1837/1836.

3. 3

Take the reciprocal of the result of step 2. The result is the “reduced mass” of the smaller body. Its unit is the same as that used in step 1.

The example above gives 1836/1837 = 0.9995. This is the reduced mass of the electron in a hydrogen atom, as compared with its original mass.

Tips and warnings

• In effect, the above calculations were the same as dividing the product of the two masses by the sum of the two masses.

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