Polar Moment of Inertia of Masses
The polar moment of inertia of a body depends on the mass of the object, its shape and its relative point of rotation
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Polar moment of inertia of masses - J - appears in dynamical equations involving rotational motion.
J has the same relationship to angular acceleration as mass has to linear acceleration.
For a point mass the polar moment of inertia is the mass times the square of perpendicular distance to the rotation reference axis and can be expressed as
J = m r2 (1)
where
J = polar moment of inertia (lbm ft2, kg m2)
m = mass (lbm, 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.
J = ∑i mi ri2 = m1 r12 + m2 r22 + ..... (2)
Convert between Moment of Inertia Units
| Multiply with | ||||||
| from | to | |||||
| kg m2 | kg cm2 | lbm ft2 | lbm in2 | slug ft2 | slug in2 | |
| kg m2 | 1 | 1 107 | 2.37 101 | 3.42 103 | 7.38 10-1 | 1.06 102 |
| kg cm2 | 1 10-7 | 1 | 2.37 10-6 | 3.42 10-4 | 7.38 10-8 | 1.06 105 |
| lbm ft2 | 4.21 10-2 | 4.21 105 | 1 | 1.44 102 | 3.11 10-2 | 4.48 |
| lbm in2 | 2.93 10-4 | 2.93 103 | 6.94 10-3 | 1 | 2.16 104 | 3.11 10-2 |
| slug ft2 | 1.36 | 1.36 107 | 3.22 101 | 4.63 103 | 1 | 1.44 102 |
| slug in2 | 9.42 10-3 | 9.42 104 | 2.23 10-1 | 3.22 101 | 6.94 10-3 | 1 |
Some Typical Bodies and their Polar Moments of Inertia
Inertia of Cylinder
Thin-walled hollow cylinder:
A thin-walled hollow cylinder is comparable with the point mass (1) and can be expressed as:
J = m r2 (3a)
where
m = mass of the hollow (lbm, kg)
r = distance between axis and the thin walled hollow (ft, m)
ro = distance between axis and outside hollow (ft, m)
Hollow cylinder:
J = 1/2 m ( ri2 + ro2) (3b)
where
m = mass of hollow (lbm, kg)
ri = distance between axis and inside hollow (ft, m)
ro = distance between axis and outside hollow (ft, m)
Solid cylinder:
J = 1/2 m r2 (3c)
where
m = mass of cylinder (lbm, kg)
r = distance between axis and outside cylinder (ft, m)
Inertia of Sphere
Thin-walled hollow sphere:
J = 2/3 m r2 (4a)
where
m = mass of sphere hollow (lbm, kg)
r = distance between axis and hollow (ft, m)
Solid sphere:
J = 2/5 m r2 (4b)
where
m = mass of sphere (lbm, kg)
r = radius in sphere (ft, m)
Moment of Inertia - General Formula
The Inertia formula may be generally expressed as
J = k m r2 (5)
where
k = inertial constant - depending on the shape of the body
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