For a rigid body rotating about a fixed principal axis, angular momentum is related to moment of inertia and angular velocity by
\[
\vec L=I\vec\omega.
\]
Here, \(I\) is the moment of inertia about the rotation axis, and \(\vec\omega\) is the angular velocity vector.
1. Scalar form
If a rigid body rotates about one fixed axis, the magnitude of angular momentum is
\[
L=I\omega.
\]
The sign or vector direction is determined by the direction of \(\vec\omega\), using the right-hand rule.
2. Vector form
In vector form,
\[
\vec L=I\vec\omega
\]
when the body rotates about a principal axis and \(I\) is the moment of inertia about that same axis. If
\[
\vec\omega=(\omega_x,\omega_y,\omega_z),
\]
then, for the simple scalar-\(I\) model used by this calculator,
\[
\vec L=(I\omega_x,I\omega_y,I\omega_z).
\]
In more advanced rigid-body dynamics, the most general relation is tensor-based:
\[
\vec L=\mathbf I\vec\omega.
\]
That is needed when the rotation is not about a principal axis.
3. Moment of inertia
Moment of inertia measures how difficult it is to change an object's rotational motion. It depends on both mass and how far the mass is distributed from the rotation axis.
| Rigid body |
Axis |
Moment of inertia |
| Solid disk / solid cylinder |
Central symmetry axis |
\(I=\frac12MR^2\) |
| Thin hoop / ring |
Central symmetry axis |
\(I=MR^2\) |
| Solid sphere |
Any diameter |
\(I=\frac25MR^2\) |
| Thin spherical shell |
Any diameter |
\(I=\frac23MR^2\) |
| Uniform rod |
Through center, perpendicular to rod |
\(I=\frac{1}{12}ML^2\) |
| Uniform rod |
Through one end, perpendicular to rod |
\(I=\frac13ML^2\) |
| Rectangular plate |
Through center, perpendicular to plate |
\(I=\frac{1}{12}M(a^2+b^2)\) |
| Point mass |
Distance \(R\) from axis |
\(I=MR^2\) |
4. Right-hand rule direction
The angular velocity vector \(\vec\omega\) points along the rotation axis. To find its direction, curl the fingers of your right hand in the direction of rotation. Your thumb points in the direction of \(\vec\omega\).
For rotation about a principal axis with positive \(I\), \(\vec L\) points in the same direction as \(\vec\omega\):
\[
\vec L \parallel \vec\omega.
\]
5. Rotational kinetic energy
Rotational kinetic energy is
\[
K_{\mathrm{rot}}=\frac12I\omega^2.
\]
This is related to angular momentum because
\[
L=I\omega.
\]
If \(I\) is constant, then angular momentum increases linearly with angular speed.
6. Link with torque
Net external torque changes angular momentum:
\[
\vec\tau_{\mathrm{net}}=\frac{d\vec L}{dt}.
\]
For a constant torque over a short interval \(\Delta t\),
\[
\Delta\vec L=\vec\tau_{\mathrm{net}}\Delta t.
\]
Therefore,
\[
\vec L_{\mathrm{new}}=\vec L_{\mathrm{old}}+\vec\tau_{\mathrm{net}}\Delta t.
\]
7. Units
Moment of inertia has units
\[
\mathrm{kg\,m^2}.
\]
Angular velocity has units
\[
\mathrm{rad/s}.
\]
Since radians are dimensionless, angular momentum has units
\[
\mathrm{kg\,m^2/s}.
\]
8. Worked example
A solid disk has moment of inertia
\[
I=0.8\ \mathrm{kg\,m^2}
\]
and angular speed
\[
\omega=12\ \mathrm{rad/s}.
\]
The angular momentum magnitude is
\[
L=I\omega.
\]
\[
L=(0.8)(12).
\]
\[
L=9.6\ \mathrm{kg\,m^2/s}.
\]
If the disk rotates counterclockwise as viewed from \(+z\), then \(\vec\omega\) points in the \(+\hat{k}\) direction, and
\[
\vec L=(0,0,9.6)\ \mathrm{kg\,m^2/s}.
\]
Key idea: for principal-axis rotation, angular momentum points along the rotation axis and has magnitude \(I\omega\).
