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Inertia tensor
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A matrix generalising I to arbitrary axes.
L = I·ωA 3×3 symmetric matrix I_ij linking L and ω: L = I·ω. Its eigenvectors are the principal axes, with eigenvalues the principal moments.
A symmetric top has two equal principal moments.
L is not parallel to ω unless along a principal axis.
In more detail
The inertia tensor is a matrix that describes a body's moment of inertia about all possible axes. It generalises the single moment of inertia used for rotation about a fixed axis.
Real-world uses
- Robotics uses inertia tensors to compute joint dynamics in arms.
- Aircraft designers compute inertia tensors to analyse handling.
- Engineers modelling a tumbling irregular object use the inertia tensor to describe how its mass is spread in every direction.
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Frequently asked questions
Why is a matrix needed?
Because angular momentum may not be parallel to angular velocity in general, so the relation needs a matrix.
Is the inertia tensor always symmetric?
Yes, it is symmetric, and it can be diagonalised along principal axes.
What is the difference between a scalar moment and the inertia tensor?
The tensor gives the full relation between the angular velocity and the angular momentum, which may not point in the same direction.
Related topics in Rotational Motion
Tops and bicycles · Rotating frames · Euler's equations · Principal axes · Torque on a dipole · Pulley with mass