Moment of Inertia


Moment of Inertia (Mass Moment of Inertia) - I -  is a measure of an object's resistance to changes in the 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


moment of inertia
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The Moment of Inertia with respect of 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 + ..... + mnrn2        (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

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)

Some Typical Bodies and their Moments of Inertia

Cylinder

Thin-walled hollow cylinder
Moments of Inertia for a thin-walled hollow cylinder is comparable with the point mass (1) and can be expressed as:
I = m r2         (3a)
where
m = mass of the hollow (kg, slugs)
r = distance between axis and the thin walled hollow (m, ft)
ro = distance between axis and outside hollow (m, ft)
Hollow cylinder
I = 1/2 m ( ri2 + ro2)         (3b)
where
m = mass of hollow (kg, slugs)
ri = distance between axis and inside hollow (m, ft)
ro = distance between axis and outside hollow (m, ft)
Solid cylinder
I = 1/2 m r2 (3c)
where
m = mass of cylinder (kg, slugs)
r = distance between axis and outside cylinder (m, ft)
Circular Disk
I = 1/2 m r2 (3d)
where
m = mass of disk (kg, slugs)
r = distance between axis and outside disk (m, ft)

Sphere

Thin-walled hollow sphere
I = 2/3 m r2         (4a)
where
m = mass of sphere hollow (kg, slugs)
r = distance between axis and hollow (m, ft)
Solid sphere
I = 2/5 m r2         (4b)
where
m = mass of sphere (kg, slugs)
r = radius in sphere (m, ft)

Rectangular Plane

Moments of Inertia for a rectangular plane with axis through center can be expressed as
I = 1/12 m (a2 + b2)        (5)
where
a, b = short and long sides
Moments of Inertia for a rectangular plane with axis along edge can be expressed as
I = 1/3 m a2      (5b)

Slender Rod

Moments of Inertia for a slender rod with axis through center can be expressed as
I = 1/12 m L2        (6)
where
L = length of rod
Moments of Inertia for a slender rod with axis through end can be expressed as
I = 1/3 m L2      (6b)

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