Moment of Inertia
Moment of Inertia (Mass Moment of Inertia) depends on the mass of the object, its shape and its relative point of rotation
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Moment of Inertia (Mass Moment of Inertia) - I - is a measure of an object's resistance to changes in rotation direction. Moment of Inertia has the same relationship to angular acceleration as mass has to linear acceleration.
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 (lb_{m} ft^{2}, kg m^{2})
m = mass (lb_{m}, kg)
r = distance between axis and rotation mass (ft, m)
That point mass relationship are 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 Moment of Inertia Units
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
The Inertia formula may be generally expressed as
I = k m r^{2} (5)
where
k = inertial constant - depending on the shape of the body
Some Typical Bodies and their Moments of Inertia
Inertia of 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 (lb_{m}, kg)
r = distance between axis and the thin walled hollow (ft, m)
r_{o} = distance between axis and outside hollow (ft, m)
Hollow cylinder:
I = 1/2 m ( r_{i}^{2} + r_{o}^{2}) (3b)
where
m = mass of hollow (lb_{m}, kg)
r_{i} = distance between axis and inside hollow (ft, m)
r_{o} = distance between axis and outside hollow (ft, m)
Solid cylinder:
I = 1/2 m r^{2} (3c)
where
m = mass of cylinder (lb_{m}, kg)
r = distance between axis and outside cylinder (ft, m)
Inertia of Sphere
Thin-walled hollow sphere:
I = 2/3 m r^{2} (4a)
where
m = mass of sphere hollow (lb_{m}, kg)
r = distance between axis and hollow (ft, m)
Solid sphere:
I = 2/5 m r^{2} (4b)
where
m = mass of sphere (lb_{m}, kg)
r = radius in sphere (ft, m)
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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