List of moments of inertia

The moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis; it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The moments of inertia of a mass have units of dimension ML2 ([mass] × [length]2). It should not be confused with the second moment of area, which has units of dimension L4 ([length]4) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia or sometimes as the angular mass.

For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases, the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. In calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and the perpendicular axis theorems.

This article considers mainly symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.

Moments of inertia

The following are scalar moments of inertia. In general, the moment of inertia is a tensor; see below.

{| class="wikitable sortable mw-collapsible"

! Description || Figure || Moment(s) of inertia || Notes

| Point mass M at a distance r from the axis of rotation.

| style="text-align:center;" |frameless|upright|class=skin-invert-image

| <math> I = M r^2</math>

|A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.

| Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles.

| style="text-align:center;" |frameless|upright|class=skin-invert-image

|<math> I = \frac{ m_1 m_2 }{ m_1 \! + \! m_2 } x^2 = \mu x^2</math>

|Both bodies are treated as point masses: dots of different size indicate the difference in masses of bodies, not in their sizes.

| Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center.

| style="text-align:center;" |frameless|upright|class=skin-invert-image

| <math>I_\mathrm{center} = \frac{1}{12} m L^2 \,\!</math>

| Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end.

| style="text-align:center;" |frameless|upright|class=skin-invert-image

| <math>I_\mathrm{end} = \frac{1}{3} m L^2 \,\!</math>

|The expression ″thin″ indicates that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube of the same mass for r1 = r2.

|Solid cylinder of radius r, height h and mass m.

| style="text-align:center;" | 170px|class=skin-invert-image

|<math>I_z = \frac{1}{2} m r^2\,\!</math>

where <math>t=\frac{r_2-r_1}{r_2}</math> is a normalized thickness ratio;

<math>I_x = I_y = \frac{1}{12} m \left(3\left(r_2^2 + r_1^2\right)+h^2\right)</math>

|The given formula is for the <math>xy</math> plane passing through the center of mass,

which coincides with the geometric center of the cylinder.

If the xy plane is at the base of the cylinder, i.e. offset by <math>d=\frac h2,</math> then by the parallel axis theorem the following formula applies:

<math>I_x = I_y = \frac{1}{12} m \left(3\left(r_2^2 + r_1^2\right)+4h^2\right)</math>

| With a density of ρ and the same geometry

| <math>I_z = \frac{\pi\rho h}{2} \left(r_2^4 - r_1^4\right)</math>

<math>I_x = I_y = \frac{\pi\rho h}{12} \left(3(r_2^4 - r_1^4)+h^2(r_2^2 - r_1^2)\right)</math>

| Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass

| style="text-align:center;" | 170px|class=skin-invert-image

| <math>I_\mathrm{solid} = \frac{1}{20} m s^2\,\!</math>

<math>I_\mathrm{hollow} = \frac{1}{12} m s^2\,\!</math> <math>I_{x, \mathrm{solid = I_{y, \mathrm{solid = I_{z, \mathrm{solid = \frac{1}{10}m s^2\,\!</math>

About an axis passing through the tip:

<math>I_x = I_y = m \left(\frac{3}{20} r^2 + \frac{3}{5} h^2\right) \,\!</math>

| Ellipsoid (solid) of semiaxes a, b, and c with mass m

| style="text-align:center;" | 170px|class=skin-invert-image

|<math>I_x = \frac{1}{5} m\left(b^2+c^2\right)\,\!</math> <math>I_y = \frac{1}{5} m \left(a^2+c^2\right)\,\!</math> <math>I_z = \frac{1}{5} m \left(a^2+b^2\right)\,\!</math>

| Thin rectangular plate of height h, width w and mass m (Axis of rotation at the end of the plate)

| style="text-align:center;" | 170px|class=skin-invert-image

|<math>I_e = \frac{1}{12} m \left(4h^2 + w^2\right)\,\!</math>

|For the purposes of moment of inertia, a rectangular plate or cuboid is equivalent to a point mass 2m/3 at the center of mass and the remaining mass spread evenly between point masses at each vertex.

| Thin rectangular plate of height h, width w and mass m (Axis of rotation at the center)

| style="text-align:center;" | 170px|class=skin-invert-image

|<math>I_c = \frac{1}{12} m \left(h^2 + w^2\right)\,\!</math>

| style="text-align:center;" | class=skin-invert-image

|<math>I_h = \frac{1}{12} m \left(w^2+d^2\right)</math> <math>I_w = \frac{1}{12} m \left(d^2+h^2\right)</math> <math>I_d = \frac{1}{12} m \left(w^2+h^2\right)</math>

|For a cube with sides <math>s</math>, <math>I = \frac{1}{6} m s^2\,\!</math>.

| Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.

| style="text-align:center;" | 140px|class=skin-invert-image

|<math>I = \frac{1}{6}m \left(\frac{W^2D^2+D^2L^2+W^2L^2}{W^2+D^2+L^2}\right)</math>

|For a cube with sides <math>s</math>, <math>I = \frac{1}{6} m s^2\,\!</math>.

| Tilted solid cuboid of depth d, width w, and length l, and mass m, rotating about the vertical axis (axis y as seen in figure).

|<math>I = \frac{m}{12} \left(l^2 \cos^2\beta + d^2 \sin^2\beta + w^2\right)</math>

|For a cube with sides <math>s</math>, <math>I = \frac{1}{6} m s^2\,\!</math>.

|Solid parallelopiped of mass m with vertices <math>\mathbf{A}_{ijk} = \mathbf{A}+i\mathbf{P} + j\mathbf{Q}+k\mathbf{R}</math> for <math>i,j,k \in \{0,1\}</math>Rotated about an axis through a point E and parallel to a unit vector L.

The centre of mass is <math>\bar{\mathbf{X = \mathbf{A}+\frac{\mathbf{P}+\mathbf{Q}+\mathbf{R{2}</math>

| style="text-align: center;" |

|<math>I_{\mathbf{E},\mathbf{L = \frac{1}{12} m \left(|\pi_\mathbf{L}\mathbf{P}|^2 + |\pi_\mathbf{L}\mathbf{Q}|^2+ |\pi_\mathbf{L}\mathbf{R}|^2\right) </math> <math>\phantom{IEL=}+ m\left|\pi_\mathbf{L}(\mathbf{E}-\bar{\mathbf{X})}\right|^2 </math>where <math>\pi_\mathbf{L} = \mathrm{I}_3 - \mathbf{L}\mathbf{L}^T</math> is the projection onto the plane perpendicular to L.

|For the purposes of moment of inertia, a parallelopiped is equivalent to a point mass 2m/3 at the center of mass and the remaining mass spread evenly between point masses at each vertex.

This result generates the above results for solid cuboids. Additionally, by letting <math>\mathbf{R} = \mathbf{0}</math> lower dimension cases are also recovered.

For a hollow parallelopiped the m/12 term is replaced by 5m/36.

| Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin.

|<math>I=\frac{1}{6}m(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}\cdot\mathbf{Q})</math>

|For the purposes of moment of inertia, a triangle is equivalent to a point mass 3m/4 at the center of mass and point masses m/12 at each vertex.

|Split the polygon into n triangles, all with the barycenter as one vertex, and the other two vertices from the polygon. Each triangle is an isosceles triangle with side length R , angle <math display="inline">\frac{2\pi}{n}</math> and mass <math display="inline">\frac{m}{n}</math>. Sum the moments for each triangle (using the previous result) to get the answer.

| Infinite disk with mass distributed in a Bivariate Gaussian distribution on two axes around the axis of rotation with mass-density as a function of the position vector <math>{\mathbf x}</math>

<math display="block">\rho({\mathbf x}) = m\frac{\exp\left(-\frac 1 2 {\mathbf x}^\mathrm{T}{\boldsymbol\Sigma}^{-1}{\mathbf x}\right)}{\sqrt{(2\pi)^2|\boldsymbol\Sigma|

</math>

| style="text-align:center;" | 130px|class=skin-invert-image

| <math>I = m \cdot \operatorname{tr}({\boldsymbol\Sigma}) \,\!</math>

<!-- There is no such thing as an illegal set of axes. They may be invalid for some purposes but the x, y and z may just be labels. The right-hand rule has no bearing here.

the x-y-z axis for the solid cylinder does not follow the right-hand rule and is an illegal set of axis. -->

For 'hollow' versions of some uniformly distributed solids, the ratio of moments of inertia is based on the dimension. More precisely, if the 'hollow' version is generated from a scaling of the solid (more precisely a homothety), in n dimensions, about a point on the axis of rotation, then the ratio of the formula for the moments of inertia is <math display="inline">\frac{n+2}{n}</math>.

To illustrate from above: the 'hollow' version of a solid disk (based on a 2 dimensional scaling about the centre) is a circle. The ratio of the moments of inertia is <math display="inline">\frac{2+2}{2} = 2</math>. A thin cylindrical shell is a 'hollow' version of a solid cylinder based on a 2 dimensional scaling (about the central axis: in x and y dimensions in the example above) and so the ratio is again 2. For the hollow and solid sphere the ratio is <math display="inline">\frac{3+2}{3} = \frac{5}{3}</math> since the hollowing is based on a 3 dimensional homothety about the centre. Regular polyhedrons have similar ratios. The hollow cone is based on a scaling about the centre of the base. So formulas based on axes that pass through the centre of the base will follow the pattern.

List of 3D inertia tensors

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

To obtain the scalar moments of inertia I above, the tensor moment of inertia I is projected along some axis defined by a unit vector n according to the formula:

<math display="block">\mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n}\equiv n_i I_{ij} n_j\,,</math>

where the dots indicate tensor contraction and the Einstein summation convention is used. In the above table, n would be the unit Cartesian basis ex, ey, ez to obtain Ix, Iy, Iz respectively.

{|class="wikitable"

! Description !! Figure !! Moment of inertia tensor

| Solid sphere of radius r and mass m

| style="background-color: whitesmoke;" | 180px

| <math>

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}

</math>

|Hollow sphere of radius r and mass m||style="background-color: whitesmoke;"|180px||

<math>

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}

</math>

| Solid ellipsoid of semi-axes a, b, c and mass m

| style="background-color: whitesmoke;" | 180px

| <math>

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}

</math>

| Right circular cone with radius r, height h and mass m, about the apex

| style="background-color: whitesmoke;" | 180px

| <math>

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}

</math>

|Slender rod along y-axis of length h and mass m about end|| center||

<math>

I =

\begin{bmatrix}

\frac{1}{3} m h^2 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & \frac{1}{3} m h^2

\end{bmatrix}

</math>

|Slender rod along y-axis of length h and mass m about center|| 180px|center||

<math>

I =

\begin{bmatrix}

\frac{1}{12} m h^2 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & \frac{1}{12} m h^2

\end{bmatrix}

</math>

|Solid rectangular plate of width w (x-direction) and height h (y-direction), and mass m

|alt=Rectangular plane with width w (x-direction) and height h (y-direction)|168x168px

|<math>

I =

\begin{bmatrix}

\frac{1}{12} m h^2 & 0 & 0 \\

0 & \frac{1}{12} m w^2 & 0 \\

0 & 0 & \frac{1}{12} m (w^2 + h^2)

\end{bmatrix}

</math>

| Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m

|| 180x|center

|| <math>

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}

</math>

|Solid cylinder of radius r, height h and mass m|| style="background-color: whitesmoke;" | 180px||

<math>

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}

</math>

|Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m|| style="background-color: whitesmoke;" | 180px||

<math>

I =

\begin{bmatrix}

\frac{1}{12} m (3(r_2^2 + r_1^2)+h^2) & 0 & 0 \\

0 & \frac{1}{12} m (3(r_2^2 + r_1^2)+h^2) & 0 \\

0 & 0 & \frac{1}{2} m (r_2^2 + r_1^2)\end{bmatrix}

</math>

Notes

References

External links

The inertia tensor of a tetrahedron
Tutorial on deriving moment of inertia for common shapes

he:טנזור התמד#דוגמאות

Moments of inertia

List of 3D inertia tensors

See also

Notes

References

External links