16.3 :  任意の軸の周りの慣性モーメント

The moment of inertia is commonly discussed in relation to principal axes, but it can also be calculated for any arbitrary axis.

When considering an arbitrary axis, the moment of inertia is determined by integrating the mass distribution of the object along that axis.

Here, the perpendicular distance between an arbitrarily chosen axis and the considered mass distribution is given by the cross-product of the unit vector that defines the direction of the axis and the position vector for the mass element.

Performing the dot product operation and expanding the brackets gives the expression for the moment of inertia.

Using the definitions of the moment of inertia and the product of inertia along the different axes, the moment of inertia along an arbitrary axis can be generalized.

If the inertia tensor is defined with respect to the xyz axes, then the moment of inertia about an arbitrary axis can be calculated if the direction cosines of the axis are known.