Calculate moment of inertia

Section 11

Moments of inertia. Radius of gyration.

It is called the moment of inertia of a system of material points in relation to a plane, an axis or a pole, the sum of the products between the masses of the particles that make up the system and the square of the distances of these particles to the plane, axis or pole considered:

Compared to a Cartesian reference system we have:

– planar moments of inertia:

– axial moments of inertia:

– polar moment of inertia:

– centrifugal moments of inertia:

The radius of gyration is the distance at which the entire mass of the material system M must be placed, concentrated at a single point on a plane, axis, or pole to obtain the same value of the planar, axial, or polar moment of inertia as and that given by the whole material system.

Properties:

– planar, axial or polar moments of inertia are positive quantities. They are null only when the system of material points is contained in the plane, on the axis or in the pole to which we refer;

– the axial moments of inertia are equal to the sum of the moments of inertia in relation to two rectangular planes:

– the polar moment of inertia can be calculated as:

• the sum of the axial moments of inertia in relation to three rectangular axes passing through that point:

• sum of planar moments of inertia:

the sum of the moments of inertia in relation to a plane and a normal axis to that plane:

– centrifugal moments of inertia can be positive, negative or zero.