m = mass (kg, slugs)
r = distance between axis and rotation mass (m, ft)
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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
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
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 |
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
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)
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)
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)
I = 1/2 m r 2 (3c)
where
m = mass of cylinder (kg, slugs)
r = distance between axis and outside cylinder (m, ft)
I = 1/2 m r 2 (3d)
where
m = mass of disk (kg, slugs)
r = distance between axis and outside disk (m, ft)
I = 2/3 m r 2 (4a)
where
m = mass of sphere hollow (kg, slugs)
r = distance between axis and hollow (m, ft)
I = 2/5 m r 2 (4b)
where
m = mass of sphere (kg, slugs)
r = radius in sphere (m, ft)
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)
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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