Formula and Method for Reduced Mass
When two bodies interact only through a mutual force — gravity between two orbiting masses, the bond between two atoms in a diatomic molecule, or an electron orbiting a nucleus — their motion can be described more simply by switching to the relative coordinate between them. Newton's second law applied to each body separately (m₁r₁'' = F and m₂r₂'' = −F) combines into a single equation, μr'' = F, where r = r₁ − r₂ is the separation between the bodies and μ is the reduced mass: μ = (m₁ × m₂) / (m₁ + m₂). This turns a two-body problem into an equivalent one-body problem of a single particle of mass μ.
How the calculation works
Enter both masses in the same unit — the unit selector only labels the output, it does not convert between m₁ and m₂, so make sure both values already use that unit consistently. The calculator multiplies the two masses and divides by their sum. An equivalent way to compute the same quantity is the reciprocal-sum form 1/μ = 1/m₁ + 1/m₂, identical to how two resistors combine in parallel. Because μ is a ratio of a product to a sum of masses expressed in the same unit, the result comes out in that same unit with no separate unit-conversion step needed.
Common mistakes
- Mixing mass units: if m₁ is entered in kilograms and m₂ in pounds, the result is meaningless — convert both to the same unit first.
- Confusing reduced mass with average or total mass: μ is always less than the lighter of the two masses, never equal to their average (m₁+m₂)/2 or their sum.
- Assuming μ approaches the heavier mass: it's the opposite — when one mass dominates, μ approaches the lighter mass, not the heavier one.
Real-world applications
- Diatomic molecule vibrations: the vibrational angular frequency of a chemical bond is ω = √(k/μ), where k is the bond's force constant and μ is the reduced mass of the two bonded atoms.
- Two-body orbital mechanics: in the gravitational two-body problem, the relative orbit of two masses is equivalent to a single body of mass μ orbiting a fixed center under a force set by the total mass m₁ + m₂.
- Atomic energy levels: the Bohr model of hydrogen-like atoms is refined by replacing the electron mass with the electron-nucleus reduced mass, which is why hydrogen, deuterium, and positronium have slightly different Rydberg constants.
- Scattering and collisions: two-body collision problems are commonly analyzed in the center-of-mass frame, where the relative motion behaves like a single particle of mass μ.