Reduced Mass Calculator (μ = m₁m₂ / (m₁ + m₂) for two-body problems)

Understanding the reduced mass calculator — concept and principle

This calculator finds the reduced mass of two bodies, μ = m₁m₂ ÷ (m₁ + m₂), which turns a two-body problem into a one-body problem. It also gives the total mass, μ relative to the lighter mass and where each body sits relative to the centre of mass. It is used in molecular spectroscopy, orbital mechanics and atomic physics.

When one partner is much heavier, the reduced mass is almost the light one: HCl vibrates like hydrogen on a wall.

How it is calculated

μ = m₁ m₂ ÷ (m₁ + m₂)

Total mass M = m₁ + m₂

Body 1's distance from the centre of mass = m₂ ÷ M of the separation

Masses can be in u, kg or any unit; μ comes out in the same unit. 1 u = 1.66053906660 × 10⁻²⁷ kg.

The principle

When two bodies interact only with each other, their motion splits into two simple parts: the centre of mass moves steadily with the total mass, and the separation between them changes as if a single body of mass μ moved around a fixed point. That makes problems like a vibrating diatomic molecule or a binary star much easier. μ is always smaller than the lighter mass and approaches it when the other mass is much heavier.

Worked example

Default inputs: m₁ = 1.008 u (hydrogen), m₂ = 35.45 u (chlorine).

μ = 1.008 × 35.45 ÷ (1.008 + 35.45) = 35.734 ÷ 36.458 = 0.980131 u.
In kilograms: 1.628 × 10⁻²⁷ kg.
Total mass = 36.458 u; μ ÷ lighter mass = 0.9724.
Hydrogen sits 0.9724 of the bond length from the centre of mass, so the centre of mass is very close to the chlorine atom.

Each point is the calculator re-run with only “Mass of body 1, m₁” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Mass of body 1, m₁	Reduced mass μ (same unit as inputs)
0.252	0.250221
0.504	0.496935
0.756	0.740214
1.01	0.980131
1.26	1.21675
1.51	1.45015
2.02	1.90752

How to read the result

For HCl the reduced mass is almost the hydrogen mass, which is why the light atom does most of the moving when the molecule vibrates. Isotope substitution changes μ and shifts vibrational frequencies, which is how isotopes are identified in spectra. For an electron and a proton, μ is about 0.9995 mₑ.

Common mistakes

Adding masses instead of combining them as a product over a sum.
Using average atomic weights when the spectrum is of a specific isotope.
Mixing units for the two masses.

Keep in mind: Use isotope masses for precise spectroscopy; standard atomic weights are averages.

Frequently asked questions

Why is the reduced mass smaller than either mass?

μ = m₁m₂/(m₁ + m₂) is always less than the smaller mass.

What is the reduced mass for equal masses?

Half of one mass.

Where is it used?

Molecular vibration and rotation, binary orbits and the hydrogen atom's energy levels.

Why does replacing H with D change spectra?

The reduced mass nearly doubles, lowering vibrational frequencies.

Sources: Michigan State University PHY321 (Fall 2017) lecture notes, "Two-body central-force problems" · CODATA 2018 recommended values (atomic mass constant), NIST

Checked 9 Nov 2026. Reduced mass μ = m₁m₂/M and the separation of the two-body kinetic energy T = ½MṘ² + ½μṙ²: Michigan State University PHY321 (Fall 2017) lecture notes, “Two-body central-force problems”; atomic mass constant 1.66053906660 × 10⁻²⁷ kg (CODATA 2018, standard value).

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