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Reduced Mass Calculator

Find the reduced mass of a two-body system.

Result

4.0000

How it is calculated

The reduced mass of two bodies is μ = m1 · m2 ÷ (m1 + m2). It has the same unit as the masses you enter, so use the same unit for both (kg, u, and so on). Both masses must be greater than zero. The result is shown with four decimals. It is always smaller than the lighter mass; when the two masses are equal it is exactly half of one of them, and when one mass is far larger it is close to the lighter one. It is the mass to use when a two-body motion is rewritten as one body moving around a fixed centre.

Worked example

Take m1 = 5 and m2 = 20. The product is 5 × 20 = 100. The sum is 5 + 20 = 25. Then 100 ÷ 25 = 4.0000. The reduced mass is 4.0000 in the same unit, which is below the lighter mass of 5, as it must be.

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Common mistakes

  • Adding the masses instead of using the product over the sum. The total mass and the reduced mass are different quantities.
  • Mixing units, such as one mass in kg and the other in grams. Convert first, because the tool cannot tell.
  • Expecting the result to lie between the two masses. It is always below the smaller one.
  • Entering zero or negative mass, which gives no result.

About this calculator

This calculator computes reduced mass, μ = m₁·m₂/(m₁+m₂), the effective single mass that lets a two-body problem — like two masses orbiting their common center of gravity, or two atoms vibrating in a diatomic molecule — be treated mathematically as a one-body problem orbiting or oscillating around a fixed point. When one mass is much larger than the other, μ approaches the smaller mass, which is why Earth's orbital mechanics can often ignore the Sun's slight wobble.

Physics and chemistry students working through classical mechanics or quantum mechanics problem sets, and researchers modeling binary star systems, molecular vibration spectra, or particle collisions use this.

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Comments (1)

  • Editorial team

    The unit stays whatever you enter, which is handy in molecular work. For carbon monoxide, with masses of 12 and 16 in atomic mass units, the tool gives about 6.8571 u. That is the mass used when treating the bond as a single vibrating body, and you would convert it to kilograms afterwards if a formula requires SI. Which system are you modelling, a molecule, a binary star, or a collision?

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