Reduced Mass Calculator
Why Reduced Mass Simplifies Two-Body Problems
The reduced mass allows a pair of interacting bodies to be modeled as a single equivalent mass moving in the relative coordinate system. Instead of solving Newton’s equations for two coupled masses, captures their shared inertia while preserving orbital period, energy, and angular momentum. The transformation is invaluable in gravitational and electrostatic systems because it converts a complex vector problem into one dimension along the line joining the bodies.
If both masses are equal, the reduced mass becomes exactly half of an individual mass. When one mass is much larger than the other, approaches the smaller mass, reflecting that the heavier object hardly moves. These limiting cases help explain why reduced mass closely matches intuitive expectations yet remains mathematically rigorous for precise physics calculations.
Formulas for Solving Any Unknown
Rearranging the defining equation lets you solve for whichever term is missing. To find when and are known, use
Swapping indices yields the equivalent expression for . The calculator implements these relationships directly and checks for physical constraints such as keeping the denominator positive.
Example Systems
The reference table shows how reduced mass behaves for common pairs of objects. Notice how the value hugs the lighter mass in systems with a strong size imbalance.
| System | m₁ (kg) | m₂ (kg) | μ (kg) |
|---|---|---|---|
| Electron–proton (hydrogen) | 9.11×10⁻³¹ | 1.67×10⁻²⁷ | 9.11×10⁻³¹ |
| Earth–Moon system | 5.97×10²⁴ | 7.35×10²² | 7.26×10²² |
| Equal masses | 5 | 5 | 2.5 |
Applications Across Physics
Reduced mass is a staple in celestial mechanics for computing orbital periods, escape velocities, and tidal effects. It also appears in spectroscopy when predicting the energy levels of diatomic molecules, where even small isotopic differences tweak and therefore the observed frequencies. In quantum mechanics, it enters the Schrödinger equation for hydrogen-like atoms, producing the precise energy levels confirmed by experiments.
For complementary tools, try the orbital period calculator to analyze two-body motion, the mass–energy equivalence calculator to convert mass into rest energy, or the mass converter if you need to switch units before solving a problem.