Moment Of Inertia Point Mass⚠ unverified
Physics / Mechanics · Compute the moment of inertia of a point mass about an axis
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mass | mass | kg | 1.0 | Mass of the particle |
| radius | radius | m | 1.0 | Perpendicular distance from the axis of rotation |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | I | kg*m^2 | Mass moment of inertia, in kilogram-metres squared (kg*m^2) |
The science & history
Understanding the Parameters
- $m$ — more mass → more rotational inertia.
- $r$ — distance to the axis (not necessarily a physical radius of a body). $I \propto r^{2}$.
- $I$ — resistance to angular acceleration about the chosen axis.
Derivation (Approaching a Proof)
Definition: $I = \int r_{\perp}^{2}\,dm$. For a single point, $I = m r^{2}$. Kinetic energy of the point in circular motion at angular speed $\omega$ is $\tfrac12 m v^{2} = \tfrac12 m (r\omega)^{2} = \tfrac12 (m r^{2})\omega^{2} = \tfrac12 I\omega^{2}$, which identifies $I$ as the factor in Rotational Kinetic Energy.
History
The point-mass definition extends by integration to continuous bodies; it is the starting definition in every rigid-body dynamics text.
Related Concepts: Moment Of Inertia Disk, Parallel Axis Theorem, Rotational Kinetic Energy, Torque, Angular Momentum
Notes: Registry calculator moment-of-inertia-point-mass (unverified). Single particle; axis
fixed in the inertial sense for the simple $I\alpha$ use.