Parallel Axis Theorem⚠ unverified
Physics / Mechanics · Shift a moment of inertia to a parallel axis
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| i_cm | icm | kg*m^2 | 1.0 | Moment of inertia about the axis through the centre of mass |
| mass | mass | kg | 1.0 | Mass of the body |
| distance | distance | m | 1.0 | Perpendicular distance between the centre-of-mass axis and the new parallel axis |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | I | kg*m^2 | Moment of inertia about the new parallel axis, in kilogram-metres squared (kg*m^2) |
The science & history
Understanding the Parameters
-
$I_{\mathrm{cm}}$ — must be about an axis parallel to the desired one and through the CM (e.g. disk $I_{\mathrm{cm}} = \tfrac12 mr^{2}$ for the central axis).
-
$m$ — total body mass.
- $d$ — shift distance; the $md^{2}$ term dominates when the axis is far from the CM.
- $I$ — always $\ge I_{\mathrm{cm}}$ for parallel axes.
Derivation (Approaching a Proof)
Write $I = \int |\mathbf{r}|^{2}\,dm$ about the new axis. Split $\mathbf{r} = \mathbf{r}_{\mathrm{cm}} + \mathbf{d}$ (vector from new origin to CM is fixed distance $d$). Cross terms vanish because the CM is the mass centroid ($\int \mathbf{r}_{\mathrm{cm}}\,dm = 0$), leaving
$$I = I_{\mathrm{cm}} + m d^{2}.$$
History
The parallel-axis (Steiner) theorem is a standard rigid-body identity used constantly in machine design and dynamics when tabulated $I_{\mathrm{cm}}$ must be moved to a shaft or pivot.
Related Concepts: Moment Of Inertia Disk, Moment Of Inertia Point Mass, Rotational Kinetic Energy, Torque
Notes: Registry calculator parallel-axis-theorem (unverified). Parallel axes only; not the
perpendicular-axis theorem for planar bodies.