Reduced Mass Calculator — Two-Body Problems
Enter the masses of two interacting bodies to find the reduced mass μ, the total mass, the mass ratio and the centre-of-mass position — used in orbital mechanics, quantum mechanics and collision physics.
Unit
μ = m₁ × m₂ / (m₁ + m₂)
- 1
Total mass (m₁ + m₂)
1 + 3 = 4 kg - 2
Product of masses (m₁ × m₂)
1 × 3 = 3 - 3
Reduced mass μ
3 ÷ 4 = 0.750000μ is always smaller than either mass alone; it equals m/2 for equal masses.
How does this calculator work?
Reduced mass μ = m₁ × m₂ / (m₁ + m₂). Always smaller than either mass; equals m/2 for equal masses; approaches the lighter mass when one is much heavier. Used in two-body orbital mechanics, diatomic molecule spectroscopy, and collision problems. Supports kg, g, atomic mass units (u) and solar masses.
Formula
How this is calculated
In a two-body problem — two gravitationally bound bodies, two colliding particles, or a diatomic molecule — the system can be reduced to a mathematically equivalent one-body problem: a single particle of mass μ (the reduced mass) moving in the combined central force field. This simplification applies to any conservative two-body interaction and greatly simplifies the equations of motion.
The reduced mass is μ = m₁m₂ / (m₁ + m₂). It is always positive and always smaller than either mass alone. For equal masses m₁ = m₂ = m, the reduced mass is exactly m/2. When one mass is much larger than the other (m₁ >> m₂), μ approaches m₂ — the lighter body, which is the physically intuitive limit: the lighter body orbits essentially unchanged while the heavy body barely moves. This is why the Earth-Moon reduced mass is very close to the Moon's mass.
The centre-of-mass position lies at a fraction m₂/(m₁+m₂) of the separation from m₁, or equivalently m₁/(m₁+m₂) from m₂. In atomic physics (e.g. the hydrogen atom), replacing the electron mass with the reduced mass accounts for the proton not being perfectly stationary and shifts spectral line energies by a tiny but measurable amount. The calculator supports kg, grams, atomic mass units (u) and solar masses (M☉).
Frequently asked questions
In a two-body gravitational problem (e.g. Earth-Moon), both bodies orbit their common centre of mass. Replacing the system with a single particle of mass μ orbiting a fixed centre of mass with the combined gravitational constant G(m₁+m₂) gives the same orbital equations, reducing a two-body problem to a tractable one-body problem.
In diatomic molecules (e.g. H₂, HCl) the vibrational and rotational energy levels depend on the reduced mass of the two bonded atoms: E ∝ 1/μ for vibrations and ∝ 1/μ for rotations. Replacing H with D (deuterium, mass ≈ 2 u) changes μ significantly and shifts the IR absorption spectrum measurably — a technique used in isotope labelling.
Almost — but not quite. The reduced mass μ = m₁m₂/(m₁+m₂) equals half the harmonic mean: μ = H(m₁,m₂)/2 where H is the harmonic mean. The harmonic mean of m₁ and m₂ is 2m₁m₂/(m₁+m₂). So reduced mass is exactly half the harmonic mean, or equivalently the "parallel combination" of the two masses (analogous to parallel resistors).
Also known as
TG we-Calculate Editorial Team. (2026). Reduced Mass Calculator — Two-Body Problems [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/reduced-mass-calculator
TG we-Calculate Editorial Team. "Reduced Mass Calculator — Two-Body Problems." TG we-Calculate. 2026. https://we-calculate.com/calculator/reduced-mass-calculator.
TG we-Calculate Editorial Team, "Reduced Mass Calculator — Two-Body Problems," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/reduced-mass-calculator
@misc{wecalculate_reduced_mass_calculator, title = {Reduced Mass Calculator — Two-Body Problems}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/reduced-mass-calculator}}, year = {2026}, note = {TG we-Calculate} }
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