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}, 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 zaxis 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
Thinwalled hollow cylinder
Moments of Inertia for a thinwalled 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
Thinwalled 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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