Reduced Mass Calculator

Reduced mass (μ) is a mathematical concept used to simplify two-body problems into an equivalent one-body problem. Instead of tracking how two objects mutually affect each other, you treat the system as a single fictitious particle with mass μ moving under the combined force. This makes it far easier to derive equations of motion, orbital paths, and oscillation frequencies.

The reduced mass formula is μ = (m₁ × m₂) / (m₁ + m₂), which is equivalent to the harmonic mean of the two masses divided by 2. It can also be written as 1/μ = 1/m₁ + 1/m₂. The reduced mass is always less than or equal to the smaller of the two masses.

When one mass (say m₂) is vastly larger than the other, the reduced mass μ approaches the smaller mass m₁. This is why in Earth–Sun orbital mechanics, the reduced mass is essentially just the Earth's mass — the Sun barely moves, so the problem simplifies to a single orbiting body.

Reduced mass is used in a wide range of fields: astronomers use it to model binary star and planet–star orbits; quantum physicists use it in the hydrogen atom model to account for the proton's finite mass; chemists use it to predict molecular vibrational frequencies from spring constants; and engineers apply it to coupled oscillator and shock absorber designs.

Total mass M = m₁ + m₂ represents the combined mass of the system, relevant for center-of-mass calculations. Reduced mass μ = (m₁ × m₂)/(m₁ + m₂) is always smaller than both individual masses and is used specifically for relative motion calculations. The ratio μ/M equals m₁m₂/(m₁+m₂)², a useful measure of how 'symmetric' the two masses are.

For two masses connected by a spring, the natural angular oscillation frequency is ω = √(k/μ), where k is the spring constant and μ is the reduced mass. Using the reduced mass instead of either individual mass gives the correct frequency for the relative oscillation between the two bodies, which is why this calculator includes an optional spring constant input.

This calculator supports kilograms (kg), grams (g), atomic mass units (amu — used in molecular and nuclear physics), pounds (lb), and slugs (slug — used in imperial engineering). All inputs and outputs use the same selected unit, so no unit conversion is applied between fields.

Yes. Select 'Calculate Mass 1 (m₁) from μ and m₂' or 'Calculate Mass 2 (m₂) from μ and m₁' in the Calculation Mode dropdown. Enter the known reduced mass (μ) and the known individual mass, and the calculator will solve for the missing mass using the rearranged formula: m₁ = (μ × m₂) / (m₂ − μ).