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Mass Moment of Inertia

The Mass Moment of Inertia vs. mass of object, it's shape and relative point of rotation - the Radius of Gyration.

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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 r 2 (1)

where

I = moment of inertia ( kg m 2 , slug ft 2 , lb f fts 2 )

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 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 m 2

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 i 2 = m 1 r 1 2 + m 2 r 2 2 + ..... + m n r n 2 (2)

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

I = ∫ r 2 dm                             (2b)

where

dm = mass of an infinitesimally small part of the body

Convert between Units for the Moment of Inertia

Moment of Inertia Units Converter
Multiply with
from to
kg m 2 g cm 2 lb m ft 2 lb m in 2 slug ft 2 slug in 2
kg m 2 1 1 10 7 2.37 10 1 3.42 10 3 7.38 10 -1 1.06 10 2
g cm 2 1 10 -7 1 2.37 10 -6 3.42 10 -4 7.38 10 -8 1.06 10 5
lb m ft 2 4.21 10 -2 4.21 10 5 1 1.44 10 2 3.11 10 -2 4.48
lb m in 2 2.93 10 -4 2.93 10 3 6.94 10 -3 1 2.16 10 -4 3.11 10 -2
slug ft 2 1.36 1.36 10 7 3.22 10 1 4.63 10 3 1 1.44 10 2
slug in 2 9.42 10 -3 9.42 10 4 2.23 10 -1 3.22 10 1 6.94 10 -3 1

Moment of Inertia - General Formula

A generic expression of the inertia equation is

I = k m r 2 (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 m 2 , slug ft 2 )

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 r 2 (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 r 2 (3c)

where

m = mass of cylinder (kg, slugs)

r = distance between axis and outside cylinder (m, ft)

Circular Disk

I = 1/2 m r 2 (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 r 2 (4a)

where

m = mass of sphere hollow (kg, slugs)

r = distance between axis and hollow (m, ft)

Solid sphere

I = 2/5 m r 2 (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 (a 2 + b 2 )                                 (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 a 2 (5b)

Slender Rod

Moments of Inertia for a slender rod with axis through center can be expressed as

I = 1/12 m L 2 (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 L 2 (6b)

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