List of moments of inertia
Updated: 5/20/2026, 7:04:01 PM Wikipedia source
The moment of inertia, denoted by I, 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.
Tables
· Moments of inertia
Point mass M at a distance r from the axis of rotation.
Point mass M at a distance r from the axis of rotation.
Description
Point mass M at a distance r from the axis of rotation.
Figure
Moment(s) of inertia
I
=
M
r
2
{\displaystyle I=Mr^{2}}
Notes
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.
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.
Description
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.
Figure
Moment(s) of inertia
I
=
m
1
m
2
m
1
Notes
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.
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center.
Description
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center.
Figure
Moment(s) of inertia
I
c
e
n
t
e
r
=
1
12
m
L
2
{\disp
Notes
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.
For the purposes of moment of inertia, the rod is equivalent to a point mass 2m/3 at the center and point masses m/6 at each end.
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end.
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end.
Description
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end.
Figure
Moment(s) of inertia
I
e
n
d
=
1
3
m
L
2
{\displaystyle I_{\mathrm {end} }={\frac {1}{3}}mL^{2}\
Notes
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.
Thin circular loop of radius r and mass m.
Thin circular loop of radius r and mass m.
Description
Thin circular loop of radius r and mass m.
Figure
Moment(s) of inertia
I
z
=
m
r
2
{\displaystyle I_{z}=mr^{2}\!}
I
x
=
I
y
Notes
This is a special case of a torus for a = 0 (see below), as well as of a thick-walled cylindrical tube with open ends, with r1 = r2 and h = 0
Thin, solid disk of radius r and mass m.
Thin, solid disk of radius r and mass m.
Description
Thin, solid disk of radius r and mass m.
Figure
Moment(s) of inertia
I
z
=
1
2
m
r
2
{\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!}
I
x
Notes
This is a special case of the solid cylinder, with h = 0. That
I
x
=
I
y
=
I
z
A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2
A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2
Description
A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2
Figure
Moment(s) of inertia
I
z
=
1
2
m
(
r
1
2
+
r
2
2
An annulus with a constant area density
ρ
A
{\displaystyle \rho _{A}}
An annulus with a constant area density
ρ
A
{\displaystyle \rho _{A}}
Description
An annulus with a constant area density
ρ
A
{\displaystyle \rho _{A}}
Figure
I
z
=
1
2
π
ρ
A
(
r
2
4
−
r
Thin cylindrical shell with open ends, of radius r and mass m.
Thin cylindrical shell with open ends, of radius r and mass m.
Description
Thin cylindrical shell with open ends, of radius r and mass m.
Figure
Moment(s) of inertia
I
=
m
r
2
{\displaystyle I=mr^{2}\,\!}
Notes
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.
Solid cylinder of radius r, height h and mass m.
Description
Solid cylinder of radius r, height h and mass m.
Figure
Moment(s) of inertia
I
z
=
1
2
m
r
2
{\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!}
I
x
Notes
This is a special case of the thick-walled cylindrical tube, with r1 = 0.
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.
Description
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.
Figure
Moment(s) of inertia
I
z
=
1
2
m
(
r
2
2
+
r
Notes
The given formula is for the
x
y
{\displaystyle 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 . offset by
d
=
h
With a density of ρ and the same geometry
With a density of ρ and the same geometry
Description
With a density of ρ and the same geometry
Figure
I
z
=
π
ρ
h
2
(
r
2
4
Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass
Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass
Description
Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass
Figure
Moment(s) of inertia
I
s
o
l
i
d
=
1
20
m
s
2
{\displaystyle I_{\mat
Notes
For the purposes of moment of inertia, a solid tetrahedron is equivalent to a point mass 4m/5 at the center of mass and point masses m/20 at each vertex.
Regular octahedron of side s and mass m
Regular octahedron of side s and mass m
Description
Regular octahedron of side s and mass m
Figure
Moment(s) of inertia
I
x
,
h
o
l
l
o
w
=
I
y
,
h
o
l
l
o
Regular dodecahedron of side s and mass m
Regular dodecahedron of side s and mass m
Description
Regular dodecahedron of side s and mass m
Moment(s) of inertia
I
x
,
h
o
l
l
o
w
=
I
y
,
h
o
l
l
o
Regular icosahedron of side s and mass m
Regular icosahedron of side s and mass m
Description
Regular icosahedron of side s and mass m
Moment(s) of inertia
I
x
,
h
o
l
l
o
w
=
I
y
,
h
o
l
l
o
Hollow sphere of radius r and mass m.
Hollow sphere of radius r and mass m.
Description
Hollow sphere of radius r and mass m.
Figure
Moment(s) of inertia
I
=
2
3
m
r
2
{\displaystyle I={\frac {2}{3}}mr^{2}\,\!}
Solid sphere (ball) of radius r and mass m.
Solid sphere (ball) of radius r and mass m.
Description
Solid sphere (ball) of radius r and mass m.
Figure
Moment(s) of inertia
I
=
2
5
m
r
2
{\displaystyle I={\frac {2}{5}}mr^{2}\,\!}
Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1.
Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1.
Description
Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1.
Figure
Moment(s) of inertia
I
=
2
5
m
⋅
r
2
5
−
r
Notes
When the cavity radius r1 = 0, the object is a solid ball (above).
When r1 becomes close to r2 the ratio
r
2
5
−
Right circular cone with radius r, height h, and mass m
Right circular cone with radius r, height h, and mass m
Description
Right circular cone with radius r, height h, and mass m
Figure
Moment(s) of inertia
I
z
=
3
10
m
r
2
{\displaystyle I_{z}={\frac {3}{10}}mr^{2}\,\!}
About an axis passing through the tip:
Right circular hollow cone with radius r, height h, and mass m
Right circular hollow cone with radius r, height h, and mass m
Description
Right circular hollow cone with radius r, height h, and mass m
Figure
Moment(s) of inertia
I
z
=
1
2
m
r
2
{\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!}
I
x
Torus with minor radius a, major radius b and mass m.
Torus with minor radius a, major radius b and mass m.
Description
Torus with minor radius a, major radius b and mass m.
Figure
Moment(s) of inertia
About an axis passing through the center and perpendicular to the diameter:
1
4
m
(
4
b
2
+
3
Ellipsoid (solid) of semiaxes a, b, and c with mass m
Ellipsoid (solid) of semiaxes a, b, and c with mass m
Description
Ellipsoid (solid) of semiaxes a, b, and c with mass m
Figure
Moment(s) of inertia
I
x
=
1
5
m
(
b
2
+
c
2
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)
Description
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)
Figure
Moment(s) of inertia
I
e
=
1
12
m
(
4
h
2
+
w
2
Notes
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)
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the center)
Description
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the center)
Figure
Moment(s) of inertia
I
c
=
1
12
m
(
h
2
+
w
2
Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r(Axis of rotation along a side of the plate)
Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r(Axis of rotation along a side of the plate)
Description
Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r(Axis of rotation along a side of the plate)
Moment(s) of inertia
I
=
1
3
m
r
2
{\displaystyle I={\frac {1}{3}}mr^{2}}
Thin parallelogram plate with mass m and sides given by vectors P and Q.
(Axis of rotation at the center and perpendicular to the plate)
Thin parallelogram plate with mass m and sides given by vectors P and Q.
(Axis of rotation at the center and perpendicular to the plate)
Description
Thin parallelogram plate with mass m and sides given by vectors P and Q.
(Axis of rotation at the center and perpendicular to the plate)
Moment(s) of inertia
I
c
=
1
12
m
(
|
P
|
2
Notes
For the purposes of moment of inertia, a parallelogram (or 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 can also be written in terms of distances from centre to the vertices:
I
c
Thin parallelogram plate with mass m with vertices at the origin, P, Q and P + Q.
(Axis of rotation at the origin and perpendicular to the plate)
Thin parallelogram plate with mass m with vertices at the origin, P, Q and P + Q.
(Axis of rotation at the origin and perpendicular to the plate)
Description
Thin parallelogram plate with mass m with vertices at the origin, P, Q and P + Q.
(Axis of rotation at the origin and perpendicular to the plate)
Moment(s) of inertia
I
=
1
6
m
(
2
P
⋅
P
+
3
P
⋅
Q
+
2
Q
⋅
Q
Notes
The moment of inertia about the vertex P + Q would be the same (by symmetry). The moment of inertia about the other two vertices would be
I
=
1
6
m
(
2
P
⋅
P
Thin parallelogram plate of mass m, perpendicular distance from side containing axis of rotation is r
(Axis of rotation along a side of the plate)
Thin parallelogram plate of mass m, perpendicular distance from side containing axis of rotation is r
(Axis of rotation along a side of the plate)
Description
Thin parallelogram plate of mass m, perpendicular distance from side containing axis of rotation is r
(Axis of rotation along a side of the plate)
Moment(s) of inertia
I
=
1
3
m
r
2
{\displaystyle I={\frac {1}{3}}mr^{2}}
Thin parallelogram plate of mass m in 3D space with vertices A, B, C and D.
Rotated about an axis through a point E and parallel to a unit vector L.
The centre of mass is
X
¯
=
Thin parallelogram plate of mass m in 3D space with vertices A, B, C and D.
Rotated about an axis through a point E and parallel to a unit vector L.
The centre of mass is
X
¯
=
Description
Thin parallelogram plate of mass m in 3D space with vertices A, B, C and D.
Rotated about an axis through a point E and parallel to a unit vector L.
The centre of mass is
X
¯
=
Moment(s) of inertia
I
E
,
L
=
m
12
(
|
π
Notes
This result can generate all the above results for thin rods (Q=0), thin rectangular plates and thin parallelogram plates.
Solid rectangular cuboid of height h, width w, and depth d, and mass m.
Solid rectangular cuboid of height h, width w, and depth d, and mass m.
Description
Solid rectangular cuboid of height h, width w, and depth d, and mass m.
Figure
Moment(s) of inertia
I
h
=
1
12
m
(
w
2
+
d
2
Notes
For a cube with sides
s
{\displaystyle s}
,
I
=
1
6
m
s
2
{\displaystyle I={\frac {1}{6}}ms^{2}\,\!}
.
Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.
Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.
Description
Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.
Figure
Moment(s) of inertia
I
=
1
6
m
(
W
2
D
2
Notes
For a cube with sides
s
{\displaystyle s}
,
I
=
1
6
m
s
2
{\displaystyle I={\frac {1}{6}}ms^{2}\,\!}
.
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).
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).
Description
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).
Figure
Moment(s) of inertia
I
=
m
12
(
l
2
cos
2
β
+
Notes
For a cube with sides
s
{\displaystyle s}
,
I
=
1
6
m
s
2
{\displaystyle I={\frac {1}{6}}ms^{2}\,\!}
.
Solid parallelopiped of mass m with vertices
A
i
j
k
=
A
+
i
P
+
j
Q
+
k
Solid parallelopiped of mass m with vertices
A
i
j
k
=
A
+
i
P
+
j
Q
+
k
Description
Solid parallelopiped of mass m with vertices
A
i
j
k
=
A
+
i
P
+
j
Q
+
k
Moment(s) of inertia
I
E
,
L
=
1
12
m
(
|
π
Notes
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
R
=
0
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.
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.
Description
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.
Moment(s) of inertia
I
=
1
6
m
(
P
⋅
P
+
P
⋅
Q
+
Q
⋅
Q
)
Notes
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. So this result can also be written as
I
=
1
12
m
(
Triangle with side lengths a, b, c, with mass m, rotating about an axis perpendicular to the plane and passing through the center of mass.
Triangle with side lengths a, b, c, with mass m, rotating about an axis perpendicular to the plane and passing through the center of mass.
Description
Triangle with side lengths a, b, c, with mass m, rotating about an axis perpendicular to the plane and passing through the center of mass.
Moment(s) of inertia
I
=
1
36
m
(
a
2
+
b
2
+
c
2
)
{\dis
Notes
These triangle results can be generated from each other using the parallel axis theorem.
This result can also be written as
I
=
1
12
m
(
r
1
2
Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
Description
Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
Figure
Moment(s) of inertia
I
=
m
∑
n
=
1
N
Q
n
Notes
Construct triangles with vertices at origin, Pn, Pn+1. Use triangle result above to find moment of inertia about the origin for each. Then take a weighted average of the moments, weighting by the area of each triangle.
An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane)
An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane)
Description
An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane)
Figure
Moment(s) of inertia
I
=
1
2
m
L
2
(
1
−
2
3
sin
Notes
Can be generated using triangle result about a vertex above.
Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter. R is the radius of the circumscribed circle.
Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter. R is the radius of the circumscribed circle.
Description
Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter. R is the radius of the circumscribed circle.
Moment(s) of inertia
I
=
1
2
m
R
2
(
1
−
2
3
sin
Notes
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
2
π
n
{\textstyle {\frac {
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
x
{\displaystyle {\mathbf {x} }}
ρ
(
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
x
{\displaystyle {\mathbf {x} }}
ρ
(
Description
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
x
{\displaystyle {\mathbf {x} }}
ρ
(
Figure
Moment(s) of inertia
I
=
m
⋅
tr
(
Σ
)
{\displaystyle I=m\cdot \operatorname {tr} ({\boldsymbol {\Sigma }})\,\!}
| Description | Figure | Moment(s) of inertia | Notes |
| Point mass M at a distance r from the axis of rotation. | | I = } | 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. | | I = m 1 m 2 m 1 | 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. | | I c e n t e r = 1 12 m L 2 {\disp | 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. For the purposes of moment of inertia, the rod is equivalent to a point mass 2m/3 at the center and point masses m/6 at each end. |
| Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end. | | I e n d = 1 3 }={\frac {1}{3}}mL^{2}\ | 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. |
| Thin circular loop of radius r and mass m. | | I z = =mr^{2}\!} I x = I y | This is a special case of a torus for a = 0 (see below), as well as of a thick-walled cylindrical tube with open ends, with r1 = r2 and h = 0 |
| Thin, solid disk of radius r and mass m. | | I z = 1 2 ={\frac {1}{2}}mr^{2}\,\!} I x | This is a special case of the solid cylinder, with h = 0. That I x = I y = I z |
| A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2 | | I z = 1 2 m ( r 1 2 + r 2 2 | |
| An annulus with a constant area density ρ } | I z = 1 2 π ρ A ( r 2 4 − r | ||
| Thin cylindrical shell with open ends, of radius r and mass m. | | I = \,\!} | 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. | | I z = 1 2 ={\frac {1}{2}}mr^{2}\,\!} I x | This is a special case of the thick-walled cylindrical tube, with r1 = 0. |
| Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m. | | I z = 1 2 m ( r 2 2 + r | 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 . offset by d = h |
| With a density of ρ and the same geometry | I z = π ρ h 2 ( r 2 4 | ||
| Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass | | I s o l i d = 1 20 m s 2 {\displaystyle I_{\mat | For the purposes of moment of inertia, a solid tetrahedron is equivalent to a point mass 4m/5 at the center of mass and point masses m/20 at each vertex. |
| Regular octahedron of side s and mass m | | I x , h o l l o w = I y , h o l l o | |
| Regular dodecahedron of side s and mass m | I x , h o l l o w = I y , h o l l o | ||
| Regular icosahedron of side s and mass m | I x , h o l l o w = I y , h o l l o | ||
| Hollow sphere of radius r and mass m. | | I = 2 3 {3}}mr^{2}\,\!} | |
| Solid sphere (ball) of radius r and mass m. | | I = 2 5 {5}}mr^{2}\,\!} | |
| Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1. | | I = 2 5 m ⋅ r 2 5 − r | When the cavity radius r1 = 0, the object is a solid ball (above). When r1 becomes close to r2 the ratio r 2 5 − |
| Right circular cone with radius r, height h, and mass m | | I z = 3 10 ={\frac {3}{10}}mr^{2}\,\!} About an axis passing through the tip: | |
| Right circular hollow cone with radius r, height h, and mass m | | I z = 1 2 ={\frac {1}{2}}mr^{2}\,\!} I x | |
| Torus with minor radius a, major radius b and mass m. | | About an axis passing through the center and perpendicular to the diameter: 1 4 m ( 4 b 2 + 3 | |
| Ellipsoid (solid) of semiaxes a, b, and c with mass m | | I x = 1 5 m ( b 2 + c 2 | |
| Thin rectangular plate of height h, width w and mass m (Axis of rotation at the end of the plate) | | I e = 1 12 m ( 4 h 2 + w 2 | 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) | | I c = 1 12 m ( h 2 + w 2 | |
| Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r(Axis of rotation along a side of the plate) | I = 1 3 {3}}mr^{2}} | ||
| Thin parallelogram plate with mass m and sides given by vectors P and Q. (Axis of rotation at the center and perpendicular to the plate) | I c = 1 12 m ( | P | 2 | For the purposes of moment of inertia, a parallelogram (or 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 can also be written in terms of distances from centre to the vertices: I c | |
| Thin parallelogram plate with mass m with vertices at the origin, P, Q and P + Q. (Axis of rotation at the origin and perpendicular to the plate) | I = 1 6 m ( 2 P ⋅ P + 3 P ⋅ Q + 2 Q ⋅ Q | The moment of inertia about the vertex P + Q would be the same (by symmetry). The moment of inertia about the other two vertices would be I = 1 6 m ( 2 P ⋅ P | |
| Thin parallelogram plate of mass m, perpendicular distance from side containing axis of rotation is r (Axis of rotation along a side of the plate) | I = 1 3 {3}}mr^{2}} | ||
| Thin parallelogram plate of mass m in 3D space with vertices A, B, C and D. Rotated about an axis through a point E and parallel to a unit vector L. The centre of mass is X ¯ = | I E , L = m 12 ( | π | This result can generate all the above results for thin rods (Q=0), thin rectangular plates and thin parallelogram plates. | |
| Solid rectangular cuboid of height h, width w, and depth d, and mass m. | | I h = 1 12 m ( w 2 + d 2 | , I = 1 6 {6}}ms^{2}\,\!} . |
| Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. | | I = 1 6 m ( W 2 D 2 | , I = 1 6 {6}}ms^{2}\,\!} . |
| 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). | | I = m 12 ( l 2 cos 2 β + | , I = 1 6 {6}}ms^{2}\,\!} . |
| Solid parallelopiped of mass m with vertices A i j k = A + i P + j Q + k | I E , L = 1 12 m ( | π | 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 R = 0 | |
| 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 = 1 6 m ( P ⋅ P + P ⋅ Q + Q ⋅ Q ) | 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. So this result can also be written as I = 1 12 m ( | |
| Triangle with side lengths a, b, c, with mass m, rotating about an axis perpendicular to the plane and passing through the center of mass. | I = 1 36 m ( a 2 + b 2 + c 2 ) {\dis | These triangle results can be generated from each other using the parallel axis theorem. This result can also be written as I = 1 12 m ( r 1 2 | |
| Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. | | I = m ∑ n = 1 N Q n | Construct triangles with vertices at origin, Pn, Pn+1. Use triangle result above to find moment of inertia about the origin for each. Then take a weighted average of the moments, weighting by the area of each triangle. |
| An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane) | | I = 1 2 m L 2 ( 1 − 2 3 sin | Can be generated using triangle result about a vertex above. |
| Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter. R is the radius of the circumscribed circle. | I = 1 2 m R 2 ( 1 − 2 3 sin | 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 2 π n {\textstyle {\frac { | |
| }} ρ ( | | I = m ⋅ tr ( Σ ) {\displaystyle I=m\cdot \operatorname {tr} ({\boldsymbol {\Sigma }})\,\!} |
· List of 3D inertia tensors
Solid sphere of radius r and mass m
Solid sphere of radius r and mass m
Description
Solid sphere of radius r and mass m
Figure
Moment of inertia tensor
I
=
[
2
5
m
r
2
Hollow sphere of radius r and mass m
Hollow sphere of radius r and mass m
Description
Hollow sphere of radius r and mass m
Figure
Moment of inertia tensor
I
=
[
2
3
m
r
2
Solid ellipsoid of semi-axes a, b, c and mass m
Solid ellipsoid of semi-axes a, b, c and mass m
Description
Solid ellipsoid of semi-axes a, b, c and mass m
Figure
Moment of inertia tensor
I
=
[
1
5
m
(
b
2
Right circular cone with radius r, height h and mass m, about the apex
Right circular cone with radius r, height h and mass m, about the apex
Description
Right circular cone with radius r, height h and mass m, about the apex
Figure
Moment of inertia tensor
I
=
[
3
5
m
h
2
Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m
Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m
Description
Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m
Figure
180x
Moment of inertia tensor
I
=
[
1
12
m
(
h
2
Slender rod along y-axis of length l and mass m about end
Slender rod along y-axis of length l and mass m about end
Description
Slender rod along y-axis of length l and mass m about end
Figure
Moment of inertia tensor
I
=
[
1
3
m
l
2
Slender rod along y-axis of length l and mass m about center
Slender rod along y-axis of length l and mass m about center
Description
Slender rod along y-axis of length l and mass m about center
Figure
Moment of inertia tensor
I
=
[
1
12
m
l
2
Solid cylinder of radius r, height h and mass m
Solid cylinder of radius r, height h and mass m
Description
Solid cylinder of radius r, height h and mass m
Figure
Moment of inertia tensor
I
=
[
1
12
m
(
3
r
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m
Description
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m
Figure
Moment of inertia tensor
I
=
[
1
12
m
(
3
(
r
| Description | Figure | Moment of inertia tensor |
| Solid sphere of radius r and mass m | | I = [ 2 5 m r 2 |
| Hollow sphere of radius r and mass m | | I = [ 2 3 m r 2 |
| Solid ellipsoid of semi-axes a, b, c and mass m | | I = [ 1 5 m ( b 2 |
| Right circular cone with radius r, height h and mass m, about the apex | | I = [ 3 5 m h 2 |
| Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m | 180x | I = [ 1 12 m ( h 2 |
| Slender rod along y-axis of length l and mass m about end | | I = [ 1 3 m l 2 |
| Slender rod along y-axis of length l and mass m about center | | I = [ 1 12 m l 2 |
| Solid cylinder of radius r, height h and mass m | | I = [ 1 12 m ( 3 r |
| Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m | | I = [ 1 12 m ( 3 ( r |
References
- Width perpendicular to the axis of rotation (side of plate); height (parallel to axis) is irrelevant.
- Physics for Scientists and Engineershttps://archive.org/details/physicsforscient02serw/page/202
- IEEE transactions on information theoryhttps://neilsloane.com/doc/Me82.pdf
- Classical Mechanics - Moment of inertia of a uniform hollow cylinder Archived 2008-02-07 at the Wayback Machine. LivePhhttp://www.livephysics.com/problems-and-answers/classical-mechanics/find-moment-of-inertia-of-a-uniform-hollow-cylinder.html
- The Mathematical Gazettehttps://doi.org/10.2307%2F3608345
- Vector Mechanics for Engineers, fourth ed
- Eric W. Weissteinhttp://scienceworld.wolfram.com/physics/MomentofInertiaRing.html
- physhttps://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/02%3A_Moments_of_Inertia/2.20%3A_Ellipses_and_Ellipsoids
- wwwhttps://www.mikipulley.co.jp/EN/Services/Tech_data/tech24.html
- A. Panagopoulos and G. Chalkiadakis. Moment of inertia of potentially tilted cuboids. Technical report, University of So
- Introduction to Classical Mechanics: With Problems and Solutions; first edition (8 January 2010)https://archive.org/details/introductiontocl00mori/page/320