Does moment of inertia tensor keeps changing if object is rotating about multiple axes

Question

Consider a plane circular disc kept in X-Y plane with Z axis passing through its centre. It is rotated about all threes axes with some angular velocities. In such a case, to find the inertia tensor, do we have to take instantaneous moment of inertia as the position of the coordinates of the body keeps changing, hence leading to different inertia tensor, each time?

Answer 1

The definition of the MMOI tensor is to convert angular velocity vector, to angular momentum vector (both vectors on the same basis vectors).

But for a rigid body, the MMOI tensor is defined for body riding basis vector (fixed to the body), such that if you know the rotational velocity vector on the body coordinates you could write

$$ \boldsymbol{L}_{\rm body} = \mathbf{I}_{\rm body\,} \boldsymbol{\omega}_{\rm body} \tag{1}$$

Each of these two vectors can be re-oriented to the common inertial frame (world basis vectors) using the local-to-world 3×3 rotation matrix $\mathbf{R}$

$$ \begin{aligned} \boldsymbol{\omega} &= \mathbf{R}\,\boldsymbol{\omega}_{\rm body} \\ \boldsymbol{L} &= \mathbf{R}\,\boldsymbol{L}_{\rm body} \\ \end{aligned} \tag{2}$$

Not the inverse rotation transformation is $\mathrm{R}^\top$ so angular momentum in the world frame is

$$ \begin{aligned} \boldsymbol{L} & = \mathbf{R} \boldsymbol{L}_{\rm body} \\ & = \mathbf{R} \mathbf{I}_{\rm body} \boldsymbol{\omega}_{\rm body} \\ & = \underbrace{\mathbf{R} \mathbf{I}_{\rm body} \mathbf{R}^\top}_{\rm I} \boldsymbol{\omega} \\ \boldsymbol{L} & = \mathbf{I}\, \boldsymbol{\omega} \end{aligned}$$

So by definition

$$ \mathbf{I} = \mathbf{R} \mathbf{I}_{\rm body} \mathbf{R}^\top$$ is the calculation of MMOI tensor on a general rotated state.