16.2:
Inertia Tensor
The inertia tensor is used to describe the distribution of mass and rotational inertia of a rigid body.
The inertia tensor is represented using a 3×3 matrix. Each element of the matrix corresponds to different moments of inertia about specific axes.
The diagonal elements of the inertia tensor matrix represent the moments of inertia about the principal axes of the body. These principal axes are the axes around which the body rotates most easily.
A smaller moment of inertia value along a particular principal axis implies that the body can be easily rotated around that axis.
Additionally, the off-diagonal elements of the inertia tensor matrix represent the product of inertia, which describes the coupling between different axes.
By choosing a unique inclination of the reference axes, the off-diagonal terms of the inertia tensor can be made zero, and the tensor is diagonalized.
The modified tensor then has only diagonal terms and are termed as the principal moments of inertia for the body computed with respect to the principal axes of inertia.
16.2:
Inertia Tensor
The concept of the inertia tensor is employed to depict the mass distribution and rotational inertia of a solid or rigid object. This tensor is expressed through a three-by-three matrix. Each component within this matrix corresponds to varying moments of inertia about specific axes.
The diagonal components of the inertia tensor matrix represent the moments of inertia concerning the principal axes of the object. These primary axes are defined as the axes where the object experiences the least resistance to rotation. If there is a smaller moment of inertia value along a certain principal axis, it indicates that the object can rotate more freely around that specific axis. Conversely, the off-diagonal components in the inertia tensor matrix symbolize the product of inertia. This essentially illustrates the interplay between different axes.
It is possible to make the off-diagonal elements of the inertia tensor zero by choosing a unique orientation of the reference axes. This action results in the tensor being diagonalized. The modified tensor then only includes diagonal terms, and these are identified as the principal moments of inertia for the object. These are calculated in relation to the principal axes of inertia.