Mass Moment of Inertia

Mass Moment of Inertia (Moment of Inertia) depends on the mass of the object, its shape and its relative point of rotation - Radius of Gyration

Mass Moment of Inertia (Moment of Inertia) - I -  is a measure of an object's resistance to change in rotation direction. Moment of Inertia has the same relationship to angular acceleration as mass has to linear acceleration.

• Moment of Inertia of a body depends on the distribution of mass in the body with respect to the axis of rotation

For a point mass the Moment of Inertia is the mass times the square of perpendicular distance to the 

rotation reference axis and can be expressed as

(I = m r²                                  (1

where

(I = moment of inertia (kg m², slug ft²

(m = mass (kg, slugs

(r = distance between axis and rotation mass (m, ft

Example - Moment of Inertia of a Single Mass

The Moment of Inertia with respect to rotation around the z-axis of a single mass of 1 kg distributed as a thin ring as indicated in the figure above, can be calculated as

I_z = (1 kg) ((1000 mm) (0.001 m/mm))²

 =1  kg m² 

Moment of Inertia - Distributed Masses

Point mass is the basis for all other moments of inertia since any object can be "built up" from a collection of point masses.

(I = ∑_i m_i r_i² = m₁ r₁² + m₂ r₂² + ..... + m_n r_n²                                      (2

For rigid bodies with continuous distribution of adjacent particles the formula is better expressed as an integral

(I = ∫ r² dm                             (2b

where

dm = mass of an infinitesimally small part of the body

Moment of Inertia - General Formula

A generic expression of the inertia equation is

(I = k m r²                                 (2c

where

k = inertial constant - depending on the shape of the body

(Radius of Gyration (in Mechanics

The Radius of Gyration is the distance from the rotation axis where a concentrated point mass equals the Moment of Inertia of the actual body. The Radius of Gyration for a body can be expressed as

(r_g = (I / m)^1/2                                (2d

where

(r_g = Radius of Gyration (m, ft

(I = Moment of inertia for the body (kg m², slug ft²

(m = mass of the body (kg, slugs