Moment of Inertia

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

Moment of Inertia (Mass Moment of Inertia) - I -  is a measure of an object's resistance to changes in a 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 a 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 (lbm ft2, kg m2)

m = mass (lbm, kg)

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

Example - Moment of Inertia of a Single Mass

moment of inertia

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

That point mass relationship are 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

Convert between Moment of Inertia Units

Multiply with
from to
kg m2 g cm2 lbm ft2 lbm in2 slug ft2 slug in2
kg m2 1 1 107 2.37 101 3.42 103 7.38 10-1 1.06 102
g cm2 1 10-7 1 2.37 10-6 3.42 10-4 7.38 10-8 1.06 105
lbm ft2 4.21 10-2 4.21 105 1 1.44 102 3.11 10-2 4.48
lbm in2 2.93 10-4 2.93 103 6.94 10-3 1 2.16 10-4 3.11 10-2
slug ft2 1.36 1.36 107 3.22 101 4.63 103 1 1.44 102
slug in2 9.42 10-3 9.42 104 2.23 10-1 3.22 101 6.94 10-3 1

Moment of Inertia - General Formula

Generic expression of the inertia equation

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, lbm ft2)

m = mass of the body (m , ft)

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 (lbm, kg)

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

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

Hollow cylinder

I = 1/2 m ( ri2 + ro2)         (3b)

where

m = mass of hollow (lbm, kg)

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

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

Solid cylinder

I = 1/2 m r2 (3c)

where

m = mass of cylinder (lbm, kg)

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

Circular Disk

I = 1/2 m r2 (3d)

where

m = mass of disk (lbm, kg)

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

Sphere

Thin-walled hollow sphere

I = 2/3 m r2         (4a)

where

m = mass of sphere hollow (lbm, kg)

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

Solid sphere

I = 2/5 m r2         (4b)

where

m = mass of sphere (lbm, kg)

r = radius in sphere (ft, m)

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)

Related Topics

  • Basics - Basic Information as SI-system, Unit converters, Physical constants
  • Dynamics - Dynamics Motion - velocity and acceleration
  • Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects
  • Statics - Loads - force and torque

Related Documents

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