Reduced Mass⚠ unverified
Physics / Mechanics · Compute the reduced mass of a two-body system
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mass1 | mass1 | kg | 1.0 | Mass of the first body |
| mass2 | mass2 | kg | 1.0 | Mass of the second body |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | μ | kg | Reduced mass of the pair, in kilograms (kg) |
The science & history
Understanding the Parameters
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$m_1$, $m_2$ — if $m_2 \gg m_1$, then $\mu \approx m_1$ (light particle orbits fixed heavy mass). If $m_1 = m_2 = m$, then $\mu = m/2$.
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$\mu$ — the inertia that multiplies relative acceleration: $\boldsymbol{F}_{12} = \mu\,\mathbf{a}_{\mathrm{rel}}$.
Derivation (Approaching a Proof)
Equations of motion: $m_1\ddot{\mathbf{r}}_1 = \mathbf{F}$, $m_2\ddot{\mathbf{r}}_2 = -\mathbf{F}$. Relative coordinate $\mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2$:
$$\ddot{\mathbf{r}} = \ddot{\mathbf{r}}_1 - \ddot{\mathbf{r}}_2 = \mathbf{F}\left(\frac{1}{m_1} + \frac{1}{m_2}\right) = \mathbf{F}\,\frac{m_1 + m_2}{m_1 m_2}.$$
Thus $\mathbf{F} = \mu\ddot{\mathbf{r}}$ with $\mu = m_1 m_2/(m_1+m_2)$. Used in orbital two-body reductions and diatomic vibration models.
History
Reduced mass is standard celestial mechanics and molecular physics; it makes the two-body Kepler problem isomorphic to a single body in a fixed $1/r$ potential.
Related Concepts: Gravitational Force, Circular Orbital Velocity, Conservation Of Momentum, Momentum
Notes: Registry calculator reduced-mass (unverified). Newtonian two-body definition.