A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a $3×3$ matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone could explain how to intuitively think of moment of inertia as a tensor.

  • 4
    $\begingroup$ Do you know what a 3×3 matrix is? $\endgroup$ – ja72 Sep 11 '14 at 17:28
  • $\begingroup$ @ja72 Ofcourse . $\endgroup$ – Žan Žurič Sep 11 '14 at 17:54
  • 3
    $\begingroup$ If you like this question you may also enjoy reading this and this Phys.SE posts. $\endgroup$ – Qmechanic Sep 11 '14 at 17:58

The moment of inertia tensor contains elements which relate one component of angular velocity to another component of angular momentum.

$$ \boldsymbol{L} = \mathbf{I}\,\boldsymbol{\omega} $$ $$ \begin{pmatrix} L_x \\ L_y \\ L_z \end{pmatrix} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{xy} & I_{yy} & -I_{yz} \\ I_{xz} & I_{yz} & I_{zz} \end{bmatrix} \begin{pmatrix} \omega_x \\ \omega_y \\ \omega_z \end{pmatrix}$$

So, for example, the element $I_{xz}$ relates the speed $\omega_x$ with the momentum $L_z$ and since it is always a symmetric tensor, the speed $\omega_z$ with the momentum $L_x$. If the case was that $I_{xz}=0$ then $L_z$ does not contain a component due to $\omega_x$ and vice versa.

Also see here for a post on how to rotate a mass moment of inertia tensor from the local (body) coordinate system to the inertial (world) coordinate system. Then look at this for the full dynamics of rigid bodies.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.