# 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 r^{2}(1)

where

I= moment of inertia (kg m,^{2}slugft^{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 thebody

### 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)