Reduced Mass Calculator

Reduced mass (mu): A single effective mass used to simplify the motion of a two-body system; for two masses it behaves like the product of the masses divided by their sum [3].

Mass 1 (selected unit): The first object's mass, entered in the unit chosen in "Mass unit (for all mass inputs)".

Mass 2 (selected unit): The second object's mass, entered in the same unit as mass 1.

Mass unit (for all mass inputs): The shared unit for every mass input and mass output on this calculator. Changing the unit changes the meaning of the numbers you enter; it does not convert old entries automatically.

Total mass (m1 + m2): The sum of the two masses.

Mass ratio (m1/m2): How many times larger mass 1 is than mass 2.

Smaller-mass approximation for mu: A quick check that uses the smaller of the two masses as an estimate for reduced mass when the other mass is much larger.

Approximation error (percent): The signed percent difference between the exact reduced mass and the smaller-mass approximation.

Two-body system: A pair of interacting objects that can often be rewritten as an equivalent one-body problem using reduced mass [4].

Diatomic molecule context: In chemistry models of two atoms moving together, reduced mass is used so the pair can be treated as one simpler system [4][4].

Mistake: Entering "Mass 1 (selected unit)" in kg and "Mass 2 (selected unit)" in g at the same time.
Fix: Use one choice in "Mass unit (for all mass inputs)" and enter every mass in that same unit.

Mistake: Filling in "Reduced mass, mu (selected unit)" while the mode is set to calculate "Reduced mass (mu)".
Fix: In reduced-mass mode, only "Mass 1 (selected unit)" and "Mass 2 (selected unit)" are used. Change the mode if you want to solve from mu.

Mistake: Trying to solve for "Mass 1 (m1)" when "Mass 2 (selected unit)" is equal to or smaller than "Reduced mass, mu (selected unit)".
Fix: Make "Mass 2 (selected unit)" greater than "Reduced mass, mu (selected unit)" before calculating "Mass 1 (m1)".

Mistake: Typing 0 or a negative value into "Mass 1 (selected unit)", "Mass 2 (selected unit)", or "Reduced mass, mu (selected unit)".
Fix: Enter a number greater than 0 in each required visible mass field.

Mistake: Thinking a large "Mass ratio (m1/m2)" means the "Reduced mass (mu)" should also be large.
Fix: Check "Smaller-mass approximation for mu". When one mass is much larger, "Reduced mass (mu)" gets close to the smaller mass, not the larger one.

Mistake: Reading "Approximation error (percent)" as always positive.
Fix: Keep the sign. A negative value means the smaller-mass approximation is larger than the exact reduced mass, which is normal for valid inputs.

• All masses must be positive real numbers. Zero, negative, blank, or non-numeric entries are invalid and should be corrected before using the result.
• The calculator assumes all visible mass inputs use the same unit chosen in "Mass unit (for all mass inputs)". It does not auto-convert existing numbers when you switch units.
• In "Mass 1 from reduced mass (mu) and mass 2" mode, the method only works when mass 2 is greater than mu. In "Mass 2 from reduced mass (mu) and mass 1" mode, the method only works when mass 1 is greater than mu.
• If the given mass equals mu in a solve-for mode, the denominator becomes zero and the missing mass would have to be infinite, so no physical result is shown.
• The smaller-mass approximation is only a quick check. It is reliable when one mass is much larger than the other, but it is not exact when the masses are similar.
• Rounding and display mode change how results look, not the underlying calculation.
• This calculator focuses on the standard two-body reduced-mass formulas. It does not model extra forces, relativistic effects, measurement uncertainty, or systems with more than two bodies.

Core idea

Reduced mass turns a two-body motion problem into an equivalent one-body problem with an effective mass [2]. This is also used in chemistry models such as diatomic rotation and vibration, where the pair can be treated more simply with reduced mass [1][4].

Formulas used

μ (reduced mass) = (m1 x m2) / (m1 + m2)

m1 = (μ x m2) / (m2 - μ)

m2 = (μ x m1) / (m1 - μ)

μ_approx = smaller of m1 and m2

approximation error (percent) = 100 x (μ - μ_approx) / μ

How the calculator applies them

If you choose reduced-mass mode, it reads "Mass 1 (selected unit)" and "Mass 2 (selected unit)" and computes mu directly. If you choose a solve-for mode, it uses mu and the one visible known mass to solve for the missing mass. Hidden fields are ignored so an old value in a hidden row cannot change the result.

The calculator keeps all math in the unit you selected. Because the formula is a ratio of masses, no unit conversion is needed as long as every mass uses the same unit.

Quick checks

For valid positive masses, the reduced mass is symmetric, so swapping m1 and m2 gives the same mu. Also, mu must be less than or equal to the smaller mass. If one mass is much larger than the other, mu gets very close to the smaller mass, which is why the approximation output is useful [4].

Worked example

Suppose "Mass 1 (selected unit)" = 2 kg and "Mass 2 (selected unit)" = 3 kg.

μ = (2 x 3) / (2 + 3) = 6 / 5 = 1.2 kg

The smaller mass is 2 kg, so the quick approximation gives 2 kg.

error = 100 x (1.2 - 2) / 1.2 = -66.6667%

That large negative percent shows the approximation is poor here because the masses are not very different.

Assumptions behind the results

The formulas assume a standard two-body model with positive masses. Real measurements can differ because of rounding, input mistakes, or because a real system may need a more detailed model than the ideal reduced-mass treatment [1][2].