**Polar Moment of Inertiaj(J)**

The M.I. of a plane about an axis perpendicular to the plane of the area is called polar moment of inertia of the area with respect to the point at which the axis intersects the plane.

Polar moment of inertia of solid body (J) = I_{xx} + I_{yy} = π/32.D^{4}

Polar moment of inertia of hollow body (J) = I_{xx} + I_{yy} = π/32.(D_{o}^{4} – D_{i}^{4})

**Section Modulus(Z) **

Section modulus is the ratio of MI about the neutral axis divided by the most distant point from the neutral axis.

Z = I/y_{max }

Section modulus for circular solid shaft (Z) = (π/64.D^{4})/D/2 = π/32.D^{3}

Section modulus for circular hollow shaft (Z) = [π/64.(D_{o}^{4} – D_{i}^{4})]/D_{O}/2 = π/32.(D_{o}^{4 }– D_{i}^{4})/D_{O}

Section modulus for rectangular section (Z) = (bd^{3}/12)/d/2 = bd^{2}/6

**Polar Modulus(Z**_{p})

It is the ratio of polar moment of inertia to outer radius.

Z_{p} = J/R

Polar Modulus of solid body (Z_{p}) = [π/32.D^{4}]/D/2 = π.D^{3}/16

Polar modulus of hollow body (Z_{p}) = [π /32.(D_{o}^{4} – D_{i}^{4})]/D/2 = [π.(D_{o}^{4 }- D_{i}^{4})/D_{o}]/16