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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.

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 m2slug ft2, lbf 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

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

 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

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

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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## Citation

• The Engineering ToolBox (2005). Mass Moment of Inertia. [online] Available at: https://www.engineeringtoolbox.com/moment-inertia-torque-d_913.html [Accessed Day Month Year].

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9.19.12