# Physics equations/Equations/Moments of inertia (large table)

 Description[1] Figure Moment(s) of inertia Point mass m at a distance r from the axis of rotation. ${\displaystyle I=mr^{2}}$ Two point masses, M and m, with reduced mass ${\displaystyle \mu }$ and separated by a distance, x. ${\displaystyle I={\frac {Mm}{M\!+\!m}}x^{2}=\mu x^{2}}$ Rod of length L and mass m (Axis of rotation at the end of the rod) ${\displaystyle I_{\mathrm {end} }={\frac {mL^{2}}{3}}\,\!}$ Rod of length L and mass m ${\displaystyle I_{\mathrm {center} }={\frac {mL^{2}}{12}}\,\!}$ Thin circular hoop of radius r and mass m ${\displaystyle I_{z}=mr^{2}\!}$${\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{2}}\,\!}$ Thin cylindrical shell with open ends, of radius r and mass m ${\displaystyle I=mr^{2}\,\!}$ Solid cylinder of radius r, height h and mass m ${\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}$${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3r^{2}+h^{2}\right)}$ Sphere (hollow) of radius r and mass m ${\displaystyle I={\frac {2mr^{2}}{3}}\,\!}$ Ball (solid) of radius r and mass m ${\displaystyle I={\frac {2mr^{2}}{5}}\,\!}$ Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate) ${\displaystyle I_{e}={\frac {mh^{2}}{3}}+{\frac {mw^{2}}{12}}\,\!}$ Solid cuboid of height h, width w, and depth d, and mass m ${\displaystyle I_{h}={\frac {1}{12}}m\left(w^{2}+d^{2}\right)}$${\displaystyle I_{w}={\frac {1}{12}}m\left(h^{2}+d^{2}\right)}$${\displaystyle I_{d}={\frac {1}{12}}m\left(h^{2}+w^{2}\right)}$