Polar Moment Of Intertia

#Physics

$\displaystyle J=\int_{0}^{r} \rho ^{2} , \mathrm{d}A$

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

Examples

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

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

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