Reduced Mass Calculator
Calculate the reduced mass of a two-body system from individual masses.
Used in orbital mechanics, molecular vibration, and quantum mechanics problems.
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
| Scenario | Reduced Mass |
|---|---|
| m₁ = m₂ = m | μ = m/2 |
| m₁ ≫ m₂ | μ ≈ m₂ |
| m₁ ≪ m₂ | μ ≈ m₁ |
| Equal partner | half the individual mass |
Where Reduced Mass Appears
| Field | Use |
|---|---|
| Orbital mechanics | Kepler’s laws for two-body orbit |
| Molecular vibration | Diatomic vibration frequency ν = (1/2π)√(k/μ) |
| Quantum mechanics | Hydrogen-like atom Schrödinger equation |
| Collision physics | Center-of-mass kinetic energy |
| Gravitational waves | Binary black hole / neutron star inspiral |
Worked Example — 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 ≈ 7.252 × 10²² kg
The reduced mass is very close to the Moon’s mass because Earth is so much heavier — this is why we usually approximate the Moon as orbiting a stationary Earth.
Worked Example — Hydrogen Molecule (H₂)
Both atoms have mass 1.008 amu, so μ = 0.504 amu = half the proton mass. This shows up in the H₂ vibrational frequency (≈4400 cm⁻¹).
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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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