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.