Polar Mass Moment of Inertia, Common Shape Equations and Calculator

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

Related: Polar Area Moment of Inertia and Section Modulus.

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.

Polar Mass Moment of Inertia of a Hollow Circular Section

Polar Mass Moment of Inertia of a Hollow Circular Section

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.

2nd moment of the strip

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
Square Section

Mass Polar Moment of Inertia Axis A-A
Mass Polar Moment of Inertia Axis B-B

Mass Polar Moment of Inertia
Cylinder

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Cylinder Calculator

Mass Polar Moment of Inertia Axis A-A

 

Mass Polar Moment of Inertia Axis B-B

Mass Polar Moment of Inertia
Hollow Cylinder

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Hollow Cylinder Calculator

Mass Polar Moment of Inertia Axis B-B

Mass Polar Moment of Inertia
Pyramid, Rectangular Base

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Pyramid, Rectangular Base Calculator

Mass Polar Moment of Inertia Axis A-A

Mass Polar Moment of Inertia Axis B-B

Through the center of gravity

Mass Polar Moment of Inertia
Sphere

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Sphere Calculator

Mass Polar Moment of Inertia Axis A-A
Mass Polar Moment of Inertia Axis B-B

Mass Polar Moment of Inertia
Spherical Sector

Mass Polar Moment of Inertia  Cylinder

Open
Polar Mass Moment of Inertia Spherical Sector Calculator

Mass Polar Moment of Inertia Axis A-A
-

Mass Polar Moment of Inertia
Spherical Segment

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Spherical Segment Calculator

Mass Polar Moment of Inertia Axis A-A
-

Mass Polar Moment of Inertia
Torus

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Torus Calculator

Mass Polar Moment of Inertia Axis A-A
Mass Polar Moment of Inertia Axis B-B

Mass Polar Moment of Inertia
Paraboloid

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Parabolid Calculator

Mass Polar Moment of Inertia Axis A-A

Mass Polar Moment of Inertia Axis B-B

Through the center of gravity

Mass Polar Moment of Inertia
Cone

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Cone Calculator

Mass Polar Moment of Inertia Axis A-A

Mass Polar Moment of Inertia Axis B-B

Through the center of gravity

Mass Polar Moment of Inertia
Frustrum of Cone

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Frustrum of Cone Calculator

Mass Polar Moment of Inertia Axis A-A
-
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

Mass Polar Moment of Inertia
Ellipsoid

Mass Polar Moment of Inertia

Open
Polar Mass Moment of Inertia Ellipsoid Calculator

Mass Polar Moment of Inertia Axis A-A
Mass Polar Moment of Inertia Axis B-B
Mass Polar Moment of Inertia Axis C-C

 

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