#### Polar Mass Moment of Inertia, Common Shape Equations and Calculator

Polar Mass Moment of Inertia About Axis A-A, Axis B-B and Axis C-C.

Mass Moments of Inertia, JM. formulas for mass moment of inertia of various solids are given below.

Example, Polar Mass Moment of Inertia of a Hollow Circular Section:

A strip of width dr on a hollow circular section, whose inner radius is r and outer radius is R.

The mass of the strip = 2πrdrρ, where ρ is the density of material. In order to get the mass of an individual section, integrate the mass of the strip from r to R.  The 2nd moment of the strip about the AA axis = 2πrdrρr2. To find the polar moment of inertia about the AA axis, integrate the 2nd moment from r to R. Note: In some many engineering examples the symbol I denotes the polar moment of inertia of masses; JM is used here to avoid confusion with moments of inertia of plane areas.

Where:

JM = Polar Mass Moment of Inertia (in-lbs-sec2, Kg-m-sec2)
h, b, l, L = Distance (in, m)
M = Mass = ρL = (lb - sec2 / in, N - Sec2 / m )
ρ = Density

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 Section Mass Polar Moment of Inertia Axis A-A JM Mass Polar Moment of Inertia Axis B-B JM Mass Polar Moment of Inertia Prism   Mass Polar Moment of Inertia Cylinder   Mass Polar Moment of Inertia Hollow Cylinder   Mass Polar Moment of Inertia Pyramid, Rectangular Base   Through the center of gravity Mass Polar Moment of Inertia Sphere   Mass Polar Moment of Inertia Spherical Sector  - Mass Polar Moment of Inertia Spherical Segment  - Mass Polar Moment of Inertia Torus   Mass Polar Moment of Inertia Paraboloid   Through the center of gravity Mass Polar Moment of Inertia Cone   Through the center of gravity Mass Polar Moment of Inertia Frustrum of Cone  -
 Mass Polar Moment of Inertia Ellipsoid Section Mass Polar Moment of Inertia Axis A-A JM Mass Polar Moment of Inertia Axis B-B JM Mass Polar Moment of Inertia Axis C-C JM     