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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