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Principal axes
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Axes along which the inertia tensor is diagonal.
Axes about which the inertia tensor is diagonal, so L ∥ ω. Symmetry axes are principal axes. A body rotates stably about the axes of largest and smallest I.
Cuboid: the three axes through the centre normal to its faces.
Rotation about the intermediate axis is unstable.
In more detail
The principal axes of a body are the axes along which the inertia tensor is diagonal. Rotation about a principal axis keeps angular momentum and angular velocity parallel.
Real-world uses
- Engineers align shafts with principal axes to reduce vibration.
- Spacecraft are designed to spin about their principal axis of greatest stability.
- A spinning stone or a rugby ball tends to spin stably about its principal axes, the directions where the inertia tensor is diagonal.
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Frequently asked questions
How are they found?
By diagonalising the inertia tensor, which gives eigenvalues as moments of inertia.
Are symmetric bodies simpler?
Yes. Their principal axes are often obvious from the symmetry.
How do I find the principal axes?
Diagonalise the inertia tensor, which gives the eigenvectors and the principal moments of inertia.
Related topics in Rotational Motion
Rotating frames · Euler's equations · Inertia tensor · Torque on a dipole · Pulley with mass · Atwood with real pulley