Mass Moment of Inertia

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, lb_f fts²)

m = mass (kg, slugs)

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

Example - Moment of Inertia of a Single Mass

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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

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 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)

• vs. Radius of Gyration in Structural Engineering

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)

r_o = distance between axis and outside hollow (m, ft)

Hollow cylinder

I = 1/2 m (r_i ²+ r_o ²)                                    (3b)

where

m = mass of hollow (kg, slugs)

r_i = distance between axis and inside hollow (m, ft)

r_o = 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)