# List of moment of inertia tensors

## List of 3D inertia tensors

This list of moment of inertia tensors is given for principal axes of each object.

Description Figure Moment of inertia tensor
Solid sphere of radius r and mass m $I = \begin{bmatrix} \frac{2}{5} m r^2 & 0 & 0 \\ 0 & \frac{2}{5} m r^2 & 0 \\ 0 & 0 & \frac{2}{5} m r^2 \end{bmatrix}$
Hollow sphere of radius r and mass m

$I = \begin{bmatrix} \frac{2}{3} m r^2 & 0 & 0 \\ 0 & \frac{2}{3} m r^2 & 0 \\ 0 & 0 & \frac{2}{3} m r^2 \end{bmatrix}$

Solid ellipsoid of semi-axes a, b, c and mass m $I = \begin{bmatrix} \frac{1}{5} m (b^2+c^2) & 0 & 0 \\ 0 & \frac{1}{5} m (a^2+c^2) & 0 \\ 0 & 0 & \frac{1}{5} m (a^2+b^2) \end{bmatrix}$
Right circular cone with radius r, height h and mass m, about the apex $I = \begin{bmatrix} \frac{3}{5} m h^2 + \frac{3}{20} m r^2 & 0 & 0 \\ 0 & \frac{3}{5} m h^2 + \frac{3}{20} m r^2 & 0 \\ 0 & 0 & \frac{3}{10} m r^2 \end{bmatrix}$
Solid cuboid of width w, height h, depth d, and mass m
$I = \begin{bmatrix} \frac{1}{12} m (h^2 + d^2) & 0 & 0 \\ 0 & \frac{1}{12} m (w^2 + d^2) & 0 \\ 0 & 0 & \frac{1}{12} m (w^2 + h^2) \end{bmatrix}$
Slender rod along y-axis of length l and mass m about end

$I = \begin{bmatrix} \frac{1}{3} m l^2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{3} m l^2 \end{bmatrix}$

Slender rod along y-axis of length l and mass m about center

$I = \begin{bmatrix} \frac{1}{12} m l^2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{12} m l^2 \end{bmatrix}$

Solid cylinder of radius r, height h and mass m

$I = \begin{bmatrix} \frac{1}{12} m (3r^2+h^2) & 0 & 0 \\ 0 & \frac{1}{12} m (3r^2+h^2) & 0 \\ 0 & 0 & \frac{1}{2} m r^2\end{bmatrix}$

Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m

$I = \begin{bmatrix} \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 & 0 \\ 0 & \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 \\ 0 & 0 & \frac{1}{2} m ({r_1}^2 + {r_2}^2)\end{bmatrix}$