Polar Moment of Intertia

$J={\int }_0^r{\rho }^2\mathrm{d}A$

• $r$ is the total radius of a rod under torsion
• $\rho$ is a variable representing distance from axis of the rod under torsion
• $A$ is the area element perpendicular to the radius vector of the rod

Examples

$J_{\text{circle}}=\frac{\pi R^4}{2}$

$J_{\text{hollow circle}}=\frac{\pi (R_{\text{outer}}^2-R_{\text{inner}}^2)}{2}$

$J_{\text{square}}=\frac{l^3}{6}$