# 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 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

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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

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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