List Of Moments Of Inertia
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Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
(which determines an object's resistance to ''linear'' acceleration). The moments of inertia of a mass have units of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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ass Ass most commonly refers to: * Buttocks (in informal American English) * Donkey or ass, ''Equus africanus asinus'' **any other member of the subgenus ''Asinus'' Ass or ASS may also refer to: Art and entertainment * ''Ass'' (album), 1973 albu ...
× engthsup>2). It should not be confused with the
second moment of area The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
, which is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia, and 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 In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
. Typically this occurs when the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. This article mainly considers 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

Following are scalar moments of inertia. In general, the moment of inertia is a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
, see below. {, class="wikitable" , - ! Description , , Figure , , Moment(s) of inertia , - , Point mass ''M'' at a distance ''r'' from the axis of rotation. 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. , align="center", , I = M r^2 , - , Two point masses, ''m''1 and ''m''2, with
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the 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. , align="center", , I = \frac{ m_1 m_2 }{ m_1 \! + \! m_2 } x^2 = \mu x^2 , - , Thin rod of length ''L'' and mass ''m'', perpendicular to the axis of rotation, rotating about its center. This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, with ''w'' = ''L'' and ''h'' = 0. , align="center", , I_\mathrm{center} = \frac{1}{12} m L^2 \,\!   , - , Thin rod of length ''L'' and mass ''m'', perpendicular to the axis of rotation, rotating about one end. This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, with ''h'' = ''L'' and ''w'' = 0. , align="center", , I_\mathrm{end} = \frac{1}{3} m L^2 \,\!   , - , Thin circular loop of radius ''r'' and mass ''m''. This is a special case of a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...
for ''a'' = 0 (see below), as well as of a thick-walled cylindrical tube with open ends, with ''r''1 = ''r''2 and ''h'' = 0. , align="center", , I_z = m r^2\!
I_x = I_y = \frac{1}{2} m r^2 \,\! , - , Thin, solid disk of radius ''r'' and mass ''m''. This is a special case of the solid cylinder, with ''h'' = 0. That I_x = I_y = \frac{I_z}{2}\, is a consequence of the perpendicular axis theorem. , align="center", , I_z = \frac{1}{2}m r^2\,\!
I_x = I_y = \frac{1}{4} m r^2\,\! , - , A uniform annulus (disk with a concentric hole) of mass ''m'', inner radius ''r''1 and outer radius ''r''2 , align="center" rowspan=2, , I_z=\frac{1}{2}m(r_1^2+r_2^2) I_x=I_y=\frac{1}{4}m(r_1^2+r_2^2) , - , An annulus with a constant area density \rho_A , I_z=\frac{1}{2}\pi\rho_A(r_2^4-r_1^4) I_x=I_y=\frac{1}{4}\pi\rho_A(r_2^4-r_1^4) , - , Thin cylindrical shell with open ends, of radius ''r'' and mass ''m''. This expression assumes that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for ''r''1 = ''r''2. Also, a point mass ''m'' at the end of a rod of length ''r'' has this same moment of inertia and the value ''r'' is called the
radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concent ...
. , align="center" , , I \approx m r^2 \,\!   , - , Solid cylinder of radius ''r'', height ''h'' and mass ''m''. This is a special case of the thick-walled cylindrical tube, with ''r''1 = 0. , align="center" , , I_z = \frac{1}{2} m r^2\,\!  
I_x = I_y = \frac{1}{12} m \left(3r^2+h^2\right) , - , Thick-walled cylindrical tube with open ends, of inner radius ''r''1, outer radius ''r''2, length ''h'' and mass ''m''. , rowspan="2" align="center" , , I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right)   
where ''t'' = (''r2 − r1'')/''r2'' is a normalized thickness ratio;
I_x = I_y = \frac{1}{12} m \left(3\left(r_2^2 + r_1^2\right)+h^2\right)
The above formula is for the xy 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 d=\frac h2, then by the parallel axis theorem the following formula applies:
I_x = I_y = \frac{1}{12} m \left(3\left(r_2^2 + r_1^2\right)+4h^2\right) , - , With a density of ''ρ'' and the same geometry , I_z = \frac{\pi\rho h}{2} \left(r_2^4 - r_1^4\right)
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) , - , Regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...
of side ''s'' and mass ''m'' , align="center" , , I_\mathrm{solid} = \frac{1}{20} m s^2\,\! I_\mathrm{hollow} = \frac{1}{12} m s^2\,\! , - , Regular
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...
of side ''s'' and mass ''m'' , align="center" , , I_{x, \mathrm{hollow=I_{y, \mathrm{hollow=I_{z, \mathrm{hollow = \frac{1}{6} m s^2\,\!
I_{x, \mathrm{solid = I_{y, \mathrm{solid = I_{z, \mathrm{solid = \frac{1}{10}m s^2\,\! , - , Regular
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentag ...
of side ''s'' and mass ''m'' , align="center" , , I_{x, \mathrm{hollow=I_{y, \mathrm{hollow=I_{z, \mathrm{hollow = \frac{39\phi+28}{90}m s^2 I_{x, \mathrm{solid=I_{y, \mathrm{solid=I_{z, \mathrm{solid = \frac{39\phi+28}{150}m s^2\,\! (where \phi=\frac{1+\sqrt{5{2}) , - , Regular
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...
of side ''s'' and mass ''m'' , align="center" , , I_{x, \mathrm{hollow=I_{y, \mathrm{hollow=I_{z, \mathrm{hollow = \frac{\phi^2}{6} m s^2 I_{x, \mathrm{solid=I_{y, \mathrm{solid=I_{z, \mathrm{solid = \frac{\phi^2}{10} m s^2 \,\! , - , Hollow
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of radius ''r'' and mass ''m''. , align="center" , , I = \frac{2}{3} m r^2\,\!   , - , Solid sphere (ball) of radius ''r'' and mass ''m''. , align="center" , , I = \frac{2}{5} m r^2\,\!   , - ,
Sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(shell) of radius ''r''2 and mass ''m'', with centered spherical cavity of radius ''r''1. When the cavity radius ''r''1 = 0, the object is a solid ball (above). When ''r''1 = ''r''2, \frac{r_2^5 - r_1^5}{r_2^3 - r_1^3}=\frac{5}{3}r_2^2, and the object is a hollow sphere. , align="center" , , I = \frac{2}{5} m\cdot\frac{r_2^5 - r_1^5}{r_2^3 - r_1^3}\,\!   , - ,
Right Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical th ...
circular cone with radius ''r'', height ''h'' and mass ''m'' , align="center" , , I_z = \frac{3}{10} mr^2 \,\!  
About an axis passing through the tip:
I_x = I_y = m \left(\frac{3}{20} r^2 + \frac{3}{5} h^2\right) \,\!  
About an axis passing through the base:
I_x = I_y = m \left(\frac{3}{20} r^2 + \frac{1}{10} h^2\right) \,\!
About an axis passing through the center of mass:
I_x = I_y = m \left(\frac{3}{20} r^2 + \frac{3}{80} h^2\right) \,\! , - ,
Right Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical th ...
circular hollow cone with radius ''r'', height ''h'' and mass ''m'' , align="center" , , I_z = \frac{1}{2} mr^2 \,\!  
I_x = I_y = \frac{1}{4}m\left(r^2 + 2h^2\right) \,\!   , - ,
Torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...
with minor radius ''a'', major radius ''b'' and mass ''m''. , align="center" , , About an axis passing through the center and perpendicular to the diameter: \frac{1}{4}m\left(4b^2 + 3a^2\right)  
About a diameter: \frac{1}{8}m\left(5a^2 + 4b^2\right)  
, - ,
Ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
(solid) of semiaxes ''a'', ''b'', and ''c'' with mass ''m'' , align="center" , , I_a = \frac{1}{5} m\left(b^2+c^2\right)\,\!

I_b = \frac{1}{5} m \left(a^2+c^2\right)\,\!

I_c = \frac{1}{5} m \left(a^2+b^2\right)\,\! , - , Thin rectangular plate of height ''h'', width ''w'' and mass ''m''
(Axis of rotation at the end of the plate) , align="center" , , I_e = \frac{1}{12} m \left(4h^2 + w^2\right)\,\! , - , Thin rectangular plate of height ''h'', width ''w'' and mass ''m''
(Axis of rotation at the center) , align="center" , , I_c = \frac{1}{12} m \left(h^2 + w^2\right)\,\!   , - , Thin rectangular plate of radius ''r'' and mass ''m'' (Axis of rotation along a side of the plate) , , I=\frac{1}{3}mr^2 , - , Solid
cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cu ...
of height ''h'', width ''w'', and depth ''d'', and mass ''m''. For a similarly oriented
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
with sides of length s, I_\mathrm{CM} = \frac{1}{6}m s^2\,\! , align="center" , , I_h = \frac{1}{12} m \left(w^2+d^2\right)

I_w = \frac{1}{12} m \left(d^2+h^2\right)

I_d = \frac{1}{12} m \left(w^2+h^2\right) , - , Solid
cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cu ...
of height ''D'', width ''W'', and length ''L'', and mass ''m'', rotating about the longest diagonal. For a cube with sides s, I = \frac{1}{6} m s^2\,\!. , align="center" , , I = \frac{1}{6}m \left(\frac{W^2D^2+D^2L^2+W^2L^2}{W^2+D^2+L^2}\right) , - , Tilted solid
cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cu ...
of depth ''d'', width ''w'', and length ''l'', and mass ''m'', rotating about the vertical axis (axis y as seen in figure). For a cube with sides s, I = \frac{1}{6} m s^2\,\!. , align="center" , , I = \frac{m}{12} \left(l^2 \cos^2\beta + d^2 \sin^2\beta + w^2\right)A. Panagopoulos and G. Chalkiadakis. Moment of inertia of potentially tilted cuboids. Technical report, University of Southampton, 2015. , - , 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. , , I=\frac{1}{6}m(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}\cdot\mathbf{Q}) , - , Plane
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
with vertices P1, P2, P3, ..., P''N'' and mass ''m'' uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. , align="center" , , I=m\left(\frac{\sum\limits_{n=1}^N\, \mathbf{P}_{n+1}\times\mathbf{P}_n\, \left(\left(\mathbf{P}_n\cdot\mathbf{P}_n\right)+\left(\mathbf{P}_{n}\cdot\mathbf{P}_{n+1}\right)+\left(\mathbf{P}_{n+1}\cdot\mathbf{P}_{n+1}\right)\right)}{6\sum\limits_{n=1}^{N}\, \mathbf{P}_{n+1}\times\mathbf{P}_n\\right) , - , Plane
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
with ''n''-vertices and mass ''m'' uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important co ...
. ''R'' is the radius of the
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
. , align="center" , , I=\frac{1}{2}mR^2\left(1 - \frac{2}{3}\sin^2\left(\tfrac{\pi}{n}\right)\right)   , - , An isosceles triangle of mass ''M'', vertex angle ''2β'' and common-side length ''L'' (axis through tip, perpendicular to plane) , align="center" , , I=\frac{1}{2} mL^2 \left(1 - \frac{2}{3}\sin^2\left(\beta\right)\right)   , - , Infinite disk with mass distributed in a
Bivariate Gaussian distribution In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. On ...
on two axes around the axis of rotation with mass-density as a function of the position vector {\mathbf x} :\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} , align="center" , , I = m \cdot \operatorname{tr}({\boldsymbol\Sigma}) \,\!


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 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...
n according to the formula: :\mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n}\equiv n_i I_{ij} n_j\,, where the dots indicate tensor contraction and the
Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
is used. In the above table, n would be the unit Cartesian basis e''x'', e''y'', e''z'' to obtain ''Ix'', ''Iy'', ''Iz'' respectively. {, class="wikitable" , - ! Description !! Figure !! Moment of inertia tensor , - , Solid
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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 An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
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 + h^2) & 0 \\ 0 & 0 & \frac{1}{12} m (w^2 + d^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 ''r''1, outer radius ''r''2, length ''h'' and mass ''m'', , , , 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} , -


See also

*
List of second moments of area The following is a list of second moments of area of some shapes. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis ...
* Parallel axis theorem * Perpendicular axis theorem


Notes


References

{{Reflist


External links


The inertia tensor of a tetrahedron
Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
Moments of inertia Physical quantities Rigid bodies Tensors Moment (physics) he:טנזור התמד#דוגמאות