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Polar Moment of Inertia

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Equations

(Eq1)     JO =  ∫ r 2 dA	polar moment of inertia of the area A with respect to the "pole" O
(Eq2)     JO = Ix + Iy	polar moment of inertia of a given area from the rectangular moments of inertia Ix and Iy of the area

Nomenclature

r	distance from O to the element of area dA

Details



An integral of great importance in problems concerning the torsion of cylindrical shafts and in problems dealing with the rotation of slabs is:

(Eq1)

JO =  ∫ r 2 dA

This integral is the polar moment of inertia of the area A with respect to the "pole" O.

The polar moment of inertia of a given area can be computed from the rectangular moments of inertia I_x and I_y of the area if these quantities are already known. Noting that r ² = x ² + y ²:

JO =

∫

r 2 dA =

∫

(x 2 + y 2) dA =

∫

y 2 dA +

∫

x 2 dA

that is:

(Eq2)

JO = Ix + Iy