What is Polar Moment of Inertia and where is it used?

Polar Moment of Inertia (J) measures a cross-section's resistance to torsion, calculated as J = Ix + Iy for the sum of moments about both axes. For circular sections, J = πd⁴/32 (solid) or π(d₀⁴ - dᵢ⁴)/32 (hollow). It is used in the torsion formula τ = Tr/J to calculate shear stress in shafts under twisting loads.