Reduced Mass Calculator

Reduced Mass

The reduced mass μ converts a two-body problem (two masses orbiting or oscillating around their common center of mass) into an equivalent one-body problem. This trick collapses two coupled equations of motion into a single equation governing the relative motion.

Formula

μ = (m₁ × m₂) / (m₁ + m₂)

Equivalent form:

1/μ = 1/m₁ + 1/m₂

The reduced mass is always smaller than either individual mass, and approaches the smaller of the two masses when one mass is much larger than the other.

Limiting Cases

Where Reduced Mass Appears

Worked Example: the Earth-Moon System

• m_Earth = 5.972 × 10²⁴ kg
• m_Moon = 7.342 × 10²² kg
• μ = (5.972 × 10²⁴ × 7.342 × 10²²) / (5.972 × 10²⁴ + 7.342 × 10²²)
• μ ≈ 7.253 × 10²² kg

The reduced mass lands within 1.2% of the Moon’s own mass because Earth is 81 times heavier. That is exactly why we get away with treating the Moon as orbiting a stationary Earth.

Worked Example: the Hydrogen Molecule (H₂)

Both atoms have mass 1.008 amu, so μ = 0.504 amu, half the mass of one hydrogen atom. This shows up in the H₂ vibrational frequency (≈4400 cm⁻¹).

A note on units. Whatever unit you put in comes straight back out, because μ is a ratio of masses multiplied by a mass. There is no conversion to get wrong. Kilograms in, kilograms out; atomic mass units in, atomic mass units out. That is why the unit selector on this page only changes the label. Mixing units between the two boxes, though, will quietly give you nonsense, so pick one and stay in it.

Center of Mass vs. Reduced Mass

These are distinct concepts:

• Center of mass: weighted average position of the two bodies.
• Reduced mass: effective inertia of the relative motion.

Both arise naturally when you split a two-body problem into center-of-mass motion plus relative motion.