Moment of Inertia:
Moment of Inertia (Mass 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,)
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
Iz = (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 mi ri2 = m1 r12 + m2 r22 + ….. + mn rn2 (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
Convert between Units for the Moment of Inertia:
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 thebody
Radius of Gyration:
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
rg = (I / m)1/2 (2d)
where
rg = Radius of Gyration (m, ft)
I = Moment of inertia for the body (kg m2, slug ft2)
m = mass of the body (kg, slugs)
