Reduced Mass Calculator finds the reduced mass of two bodies in kg, g, lb or u, or of a diatomic molecule from its isotopes, with the vibration frequency and the moment of inertia.
Enter the two masses. Each one can use its own unit.
Spring, Separation and Result Unit
Step-by-Step Working
- Known values: m₁ = 2 kg, m₂ = 6 kg, k = 150 N/m, r = 0.8 m
- Reduced mass: μ = m₁ × m₂ ÷ (m₁ + m₂) = 2 × 6 ÷ 8 = 1.5 kg
- Oscillation: f = √(k ÷ μ) ÷ 2π = √(150 ÷ 1.5) ÷ 2π = 1.59155 Hz
- Moment of inertia: I = μ × r² = 1.5 kg × (0.8 m)² = 0.96 kg·m²
Reduced Mass as m₂ Changes
| m₂ (kg) | μ (kg) | μ ÷ m₁ |
|---|---|---|
| 0.6 | 0.461538 | 23.0769% |
| 3 | 1.2 | 60% |
| 6 | 1.5 | 75% |
| 60 | 1.93548 | 96.7742% |
| 600 | 1.99336 | 99.6678% |
Reduced Mass Calculator for Two-Body Systems
Reduced mass is the single effective mass that stands in for two bodies moving about each other. It turns a two-body problem into a one-body problem. Two atoms on a bond, two stars in orbit and two blocks on a spring can each be treated as one body of mass μ.
The Reduced Mass Calculator has three settings under Calculate. The first finds μ from two masses. The second finds μ of a diatomic molecule from a list of isotopes. The third works backward and finds a missing mass m₂ from μ and m₁.
Each mass has its own unit menu, so a problem can mix units. The choices are kilograms, grams, pounds, atomic mass units, electron masses, Earth masses and solar masses. The result can be shown in any of them.
The starting entries are 2 kg and 6 kg. The main result reads 1.5 kg for the Two Bodies. Four cards under it give the same value in other units, compare it with the lighter mass, and add an oscillation frequency and a moment of inertia.
Reduced Mass Formula
To calculate reduced mass, multiply the two masses and divide by their sum. The symbol for it is the Greek letter μ. OpenStax University Physics gives the formula in this form for a diatomic molecule.
$$\mu = \frac{m_{1} m_{2}}{m_{1} + m_{2}}$$
For 2 kg and 6 kg that is 2 × 6 ÷ 8 = 1.5 kg. The same formula can be written with reciprocals, which shows why μ is always smaller than either mass. The two masses can be swapped without changing the answer.
$$\frac{1}{\mu} = \frac{1}{m_{1}} + \frac{1}{m_{2}}$$
The second card of the Reduced Mass Calculator shows how close μ comes to the lighter mass. In the example it is 75% of the lighter mass, 0.5 kg below it. When the masses are equal, μ is exactly half of either one.
| m₂ with m₁ fixed at 2 kg | Reduced mass μ (kg) | μ as a share of m₁ |
|---|---|---|
| 0.6 kg | 0.4615 | 23.08% |
| 3 kg | 1.2 | 60% |
| 6 kg | 1.5 | 75% |
| 60 kg | 1.935 | 96.77% |
| 600 kg | 1.993 | 99.67% |
These rows are the Reduced Mass as m₂ Changes panel. As the partner gets heavier, μ climbs toward the lighter mass and never passes it. A very heavy partner hardly moves, so the light body does nearly all the moving.
How the Reduced Mass Formula Is Derived
Take two bodies that push or pull on each other with a force F. By Newton’s third law the forces are equal and opposite, so the accelerations are F ÷ m₁ and −F ÷ m₂. The acceleration of one body relative to the other is the difference.
$$a_{rel} = \frac{F}{m_{1}} + \frac{F}{m_{2}} = \frac{F}{\mu}$$
So the separation between the bodies changes as if one body of mass μ felt the force F. That is the whole derivation. The Newton’s Third Law Calculator works through the equal and opposite pair that it starts from.
When to Use Reduced Mass
Use reduced mass when exactly two bodies interact and you care about their motion relative to each other. It applies to orbits, molecular vibration and rotation, two masses on a spring, and collisions seen from the center of mass.
It is not needed when one body is fixed, such as a mass on a spring bolted to a wall. It also does not extend to three or more bodies. For the motion of the pair as a whole, use the total mass at the center of mass.
Finding a Missing Mass from μ
The third setting of the Reduced Mass Calculator rearranges the formula for the second mass. It needs μ and one mass, and μ must be the smaller of the two numbers entered.
$$m_{2} = \frac{\mu \, m_{1}}{m_{1} – \mu}$$
With μ = 1.2 kg and m₁ = 2 kg, the missing mass is 1.2 × 2 ÷ 0.8 = 3 kg. The answer climbs steeply as μ gets close to m₁, since that only happens with a very heavy partner.
Reduced Mass Units and Values in kg
The unit of reduced mass is the unit of mass. In SI that is the kilogram. Chemistry tables list it in atomic mass units, written u or amu. To calculate reduced mass in kg from a value in u, multiply by the atomic mass constant.
The amu to kg factor is 1 u = 1.66053906892 × 10⁻²⁷ kg in the NIST 2022 CODATA values. The electron mass in the same table is 9.1093837139 × 10⁻³¹ kg. The Reduced Mass Calculator uses both figures.
| Two-body system | Reduced mass | In kg |
|---|---|---|
| Electron and proton (hydrogen atom) | 0.99946 mₑ | 9.1044 × 10⁻³¹ |
| H₂ (two ¹H atoms) | 0.5039 u | 8.368 × 10⁻²⁸ |
| ¹H³⁵Cl | 0.9796 u | 1.6267 × 10⁻²⁷ |
| ¹²C¹⁶O | 6.856 u | 1.1385 × 10⁻²⁶ |
| Earth and Moon | 98.78% of the Moon | 7.257 × 10²² |
| Earth and Sun | 99.9997% of Earth | 5.9722 × 10²⁴ |
The hydrogen atom row explains a common shortcut. The proton is 1,836 times heavier than the electron, so μ is only 0.054% below the electron mass. Using the plain electron mass is nearly right, and using μ is exactly right.
Reduced Mass in Chemistry
In chemistry, reduced mass is the effective mass of a vibrating or rotating diatomic molecule. Both atoms move, each about the shared center of mass. The bond behaves like a spring with a single mass μ on it.
The Reduced Mass Calculator has a chemistry setting, μ of a diatomic molecule. It lists 21 isotopes from hydrogen-1 to iodine-127 with their exact atomic masses. For average atomic weights, enter them in u under the two-mass setting.
How to Calculate the Reduced Mass of HCl
Pick ¹H and ³⁵Cl. Their masses are 1.007825 u and 34.968853 u. The reduced mass is 1.007825 × 34.968853 ÷ 35.976678 = 0.9796 u. In kilograms that is 0.9796 × 1.66054 × 10⁻²⁷ = 1.6267 × 10⁻²⁷ kg.
That is 97.2% of the mass of the hydrogen atom. In a hydride the light atom does almost all the moving, so μ stays just below 1 u. With average atomic weights of 1.008 and 35.45, the result is 0.9801 u.
Vibration and Rotation from μ
Under Bond Constants and Result Unit, the Reduced Mass Calculator takes a bond force constant k and a bond length r. It then gives the stretching frequency of the bond as a harmonic spring, and the rotational constant of the molecule as a rigid rotor.
$$\nu = \frac{1}{2\pi} \sqrt{\frac{k}{\mu}}$$
$$B = \frac{h}{8 \pi^{2} c \, \mu r^{2}}$$
With the starting values of 516 N/m and 127.46 pm, ¹H³⁵Cl gives 2,990 cm⁻¹ and B = 10.59 cm⁻¹. The NIST Chemistry WebBook lists 2,990.95 cm⁻¹ and 10.5934 cm⁻¹ for the same molecule, with a bond length of 1.2746 Å.
| Molecule | μ (u) | Vibration (cm⁻¹) | B (cm⁻¹) |
|---|---|---|---|
| ¹H³⁵Cl | 0.9796 | 2,990 | 10.59 |
| ¹H³⁷Cl | 0.9811 | 2,988 | 10.58 |
| ²H³⁵Cl | 1.9044 | 2,144 | 5.449 |
| ²H³⁷Cl | 1.9100 | 2,141 | 5.433 |
The Other Isotopes of the Same Elements panel builds this comparison. Swapping hydrogen for deuterium nearly doubles μ and drops the band from 2,990 to 2,144 cm⁻¹. Swapping the chlorine isotope shifts it by only about 2 cm⁻¹.
Reduced Mass in Physics: Springs and Orbits
Two masses joined by a spring swing as one mass μ on that spring. The third card gives the result for the spring constant you enter. For 2 kg and 6 kg on a 150 N/m spring, the angular frequency is √(150 ÷ 1.5) = 10 rad/s.
That is 1.592 Hz with a period of 0.6283 s. A lighter μ or a stiffer spring gives a faster swing. The spring constant comes from Hooke’s law, and the Hooke’s Law Calculator finds it from a force and a stretch.
The fourth card of the Reduced Mass Calculator covers rotation. Two bodies turning about their shared center of mass have a moment of inertia of μ × r², where r is the full separation. For the example at 0.8 m apart, that is 1.5 × 0.64 = 0.96 kg·m².
The same card places the center of mass. It sits 0.6 m from the 2 kg body and 0.2 m from the 6 kg body, closer to the heavier one. The Car Center of Mass Calculator applies that balance rule to a car on scales.
In an orbit, reduced mass lets one body circle a fixed point under the full gravitational pull between the pair. The Gravitational Force Calculator gives that pull. For the Earth and Sun, μ differs from Earth’s mass by only 3 parts in a million.
Mass Basics: Units, Types and Weight
The SI unit of mass is the kilogram, and 1 kg in mass is 1,000 grams. It is one of the 7 basic SI units listed by NIST. The others are the meter, second, mole, ampere, kelvin and candela.
Why is 1 kg exactly 1 kg? Since the 2019 revision of the SI, the kilogram is fixed by a constant of nature. The Planck constant is set at exactly 6.62607015 × 10⁻³⁴ J·s, and the kilogram follows from it together with the meter and the second.
There are two types of mass in physics. Inertial mass measures how hard a body is to accelerate. Gravitational mass measures how strongly gravity acts on it. A list of three types splits the second into active and passive gravitational mass. All of them have the same value in every test so far.
Is 50 kg or 70 kg a Mass or a Weight?
It is a mass. Kilograms measure the amount of matter, which is the same anywhere. Weight is the force of gravity on that mass, measured in newtons. A 50 kg person weighs 490.3 N on Earth, and a 70 kg person weighs 686.5 N.
On the Moon the 70 kg person still has a mass of 70 kg but weighs only about 113 N, because gravity there is about one sixth as strong. The kg to Newtons Calculator converts any mass to its weight on Earth.
What Is the Formula for Mass?
Mass is calculated in three common ways. From a force and an acceleration, m = F ÷ a. From a weight, m = W ÷ g, with g = 9.80665 m/s². From a density and a volume, m = ρ × V, which is the formula behind a density calculator.
The Newton’s Second Law Calculator solves the first form. To calculate mass in kg, keep the force in newtons, the acceleration in m/s², the density in kg/m³ and the volume in m³.
| 10 examples of mass | Mass in kg |
|---|---|
| Electron | 9.109 × 10⁻³¹ |
| Proton | 1.673 × 10⁻²⁷ |
| One atomic mass unit | 1.6605 × 10⁻²⁷ |
| Water molecule (18.015 u) | 2.99 × 10⁻²⁶ |
| One gram | 0.001 |
| One pound | 0.4536 |
| One liter of water | 1 |
| One metric ton | 1,000 |
| Earth | 5.9722 × 10²⁴ |
| Sun | 1.98841 × 10³⁰ |
The first five rows of the table are small-scale examples of mass, and the last five are everyday and astronomical ones. The Reduced Mass Calculator accepts any pair of them, since each field has its own unit.
Reduced Price and Percent Reduction
“Reduced” also turns up in price questions, which use a different formula. To calculate a reduced price, multiply the original by one minus the percentage as a decimal. To find the reduction itself, multiply the original by the percentage.
$$\text{reduced value} = \text{original} \times \left(1 – \frac{p}{100}\right)$$
| Reduction on 250 | Amount taken off | Reduced value |
|---|---|---|
| 2% | 5 | 245 |
| 6% | 15 | 235 |
| 10% | 25 | 225 |
| 20% | 50 | 200 |
| 40% | 100 | 150 |
| 70% | 175 | 75 |
| 90% | 225 | 25 |
A 20% reduction of 80 leaves 64. A plain percentage is a single multiplication, so 60% of 160 is 96, 60% of 75 is 45 and 70% of 300 is 210. A 600% reduction of 600 is not possible for a price, because anything over 100% takes it below zero.
None of this uses reduced mass, which is about two bodies moving around each other and has nothing to do with a discount. More mechanics tools are listed under Dynamics and Statics Calculators.