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| name = Moment of inertia
| image = Маховик.jpg
| caption = [[Flywheel]]s have large moments of inertia to smooth out changes in rates of rotational motion.
| unit =
| otherunits = lbf·ft·s<sup>2</sup>
| symbols = ''I''
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| image1 = Chinese Shenyang J-11 from below in August 2014.JPG
| image2 = Dg800.jpg
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The '''moment of inertia''', otherwise known as the '''mass moment of inertia''', '''angular/rotational mass''', '''second moment of mass''', or most accurately, '''rotational inertia''', of a [[rigid body]] is defined relative to a
It is an [[intensive and extensive properties|extensive]] (additive) property: for a [[point particle|point mass]] the moment of inertia is simply the mass times the square of the [[perpendicular distance]] to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second [[Moment (
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a [[Scalar (physics)|scalar]] value, matters. For bodies free to rotate in three dimensions, their moments can be described by a [[Symmetric matrix|symmetric]] 3
In [[mechanical engineering]], simply "inertia" is often used to refer to "[[Inertial Mass|inertial mass]]" or "moment of inertia".<ref>{{Cite book |last1=Escudier |first1=Marcel |url=http://www.oxfordreference.com/view/10.1093/acref/9780198832102.001.0001/acref-9780198832102 |title=A Dictionary of Mechanical Engineering |last2=Atkins |first2=Tony |date=2019 |publisher=Oxford University Press |isbn=978-0-19-883210-2 |edition=2 |language=en |doi=10.1093/acref/9780198832102.001.0001}}</ref>
==Introduction==
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The moment of inertia plays the role in rotational kinetics that [[mass]] (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by <math>mr^2</math>, where <math>r</math> is the distance of the point from the axis, and <math>m</math> is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
In 1673, [[Christiaan Huygens]] introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a [[compound pendulum]].<ref name="mach">{{cite book |last=Mach |first=Ernst |title=The Science of Mechanics |year=1919 |pages=[https://archive.org/details/scienceofmechani005860mbp/page/n196 173]–187 |url=https://archive.org/details/scienceofmechani005860mbp |access-date=November 21, 2014}}</ref> The term ''moment of inertia'' ("momentum inertiae" in [[Latin]]) was introduced by [[Leonhard Euler]] in his book ''Theoria motus corporum solidorum seu rigidorum'' in 1765,<ref name="mach"/><ref name="Euler1730">{{Cite book |last=Euler |first=Leonhard |title=Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all motions which can occur in such bodies.]|publisher= A. F. Röse|location= Rostock and Greifswald (Germany)|date= 1765|page= [https://archive.org/details/theoriamotuscor00eulegoog/page/n202 166]|url= https://archive.org/details/theoriamotuscor00eulegoog|language=la |isbn=978-1-4297-4281-8}} From page 166: ''"Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur."'' (Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)</ref> and it is incorporated into [[Euler's laws#Euler's second law|Euler's second law]].
The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.<ref name="Marion 1995">{{cite book |last1=Marion |first1=JB |last2=Thornton |first2=ST |year=1995 |title=Classical dynamics of particles & systems |edition=4th |publisher=Thomson |isbn=0-03-097302-3 |url-access=registration |url=https://archive.org/details/classicaldynamic00mari_0 }}</ref><ref name="Symon 1971">{{cite book |last=Symon |first=KR |year=1971 |title=Mechanics |edition=3rd |publisher=Addison-Wesley |isbn=0-201-07392-7}}</ref>
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== Examples ==
{{See also|List of moments of inertia}}
=== Simple pendulum ===
Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle <math>m</math> with the square of its distance <math>r</math> to the pivot, that is
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| isbn=9780471216438
| edition=7th
}}</ref>{{rp|pp=395–396}}<ref>
{{cite book
| last=French
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| location=Boca Raton, FL
| isbn=9780748744473
}}</ref>{{rp|pp=51–53}} The [[resonance|natural]] [[angular frequency|frequency]] (<math>\omega_\text{n}</math>) of a compound pendulum depends on its moment of inertia, <math>I_P</math>,
<math display="block">\omega_\text{n} = \sqrt{\frac{mgr}{I_P}},</math>
where <math>m</math> is the mass of the object, <math>g</math> is local acceleration of gravity, and <math>r</math> is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.<ref name="Uicker"/>{{rp|pp=516–517}}
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point <math>P</math> so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (<math>t</math>), to obtain
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==Measuring moment of inertia==
The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar [[pendulum]]. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.<ref>
== Moment of
Moment of
These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the
Height (''h'') and breadth (''b'') are the linear measures, except for circles, which are effectively half-breadth derived, <math>r</math>
=== Sectional areas moment calculated thus<ref>{{cite book |last1=Morrow |first1=H. W. |last2=Kokernak |first2=Robert |title=Statics and Strengths of Materials |date=2011 |publisher=Prentice Hall |location=New Jersey |isbn=
# Square: <math>I_{xx}=I_{yy}=\frac{b^4}{12}</math><br>▼
#
#
#
== Motion in a fixed plane ==
=== Point mass ===
[[File:Rolling Racers - Moment of inertia.gif|thumb|right|Four objects with identical masses and radii racing down a plane while rolling without slipping. {{paragraph}}From back to front: {{unbulleted list
| {{
| {{
| {{
| {{
}}{{paragraph}} The time for each object to reach the finishing line depends on their moment of inertia. ([[:File:Rolling Racers - Moment of inertia.ogv|OGV version]])]]
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|year=2010
|isbn=978-0195371239
}}</ref>{{rp|pp=516–517}}<ref name="Beer"/>{{rp|pp=1084–1085}}<ref name="Beer">{{cite book|author=Ferdinand P. Beer | author2=E. Russell Johnston, Jr.|author3=
<math display="block">
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Here, the [[function (mathematics)|function]] <math>\rho</math> gives the mass density at each point <math>(x, y, z)</math>, <math>\mathbf{r}</math> is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point <math>(x, y, z)</math> in the solid, and the integration is evaluated over the volume <math>V</math> of the body <math>Q</math>. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
'''Note on second moment of area''': The moment of inertia of a body moving in a plane and the [[second moment of area]] of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the <math>z</math>-axis perpendicular to the cross-section, weighted by its density. This is also called the ''polar moment of the area'', and is the sum of the second moments about the <math>x</math>- and <math>y</math>-axes.<ref>Walter D. Pilkey, [https://books.google.com/books?id=4hEsqvplmFMC&
==== Examples ====
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The moment of inertia of a '''compound pendulum''' constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.<ref name="Beer"/>
* The moment of inertia of a '''thin rod''' with constant cross-section <math>s</math> and density <math>\rho</math> and with length <math>\ell</math> about a perpendicular axis through its center of mass is determined by integration.<ref name="Beer"/>{{rp|p=1301}} Align the <math>x</math>-axis with the rod and locate the origin its center of mass at the center of the rod, then <math display="block">
I_{C, \text{rod}} = \iiint_Q \rho\,x^2 \, dV =
\int_{-\frac{\ell}{2}}^\frac{\ell}{2} \rho\,x^2 s\, dx =
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\frac{m\ell^2}{12},
</math> where <math>m = \rho s \ell</math> is the mass of the rod.
* The moment of inertia of a '''thin disc''' of constant thickness <math>s</math>, radius <math>R</math>, and density <math>\rho</math> about an axis through its center and perpendicular to its face (parallel to its axis of [[rotational symmetry]]) is determined by integration.<ref name="Beer"/>{{rp|p=1301}}{{Failed verification|date=June 2019|reason=page 1301 is the index of the book. I assume someone made a mistake with the page number}} Align the <math>z</math>-axis with the axis of the disc and define a volume element as <math>dV = sr \, dr\, d\theta</math>, then <math display="block">
I_{C, \text{disc}} = \iiint_Q \rho \, r^2\, dV =
\int_0^{2\pi} \int_0^R \rho r^2 s r\, dr\, d\theta =
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[[File:Moment of inertia solid sphere.svg|right|thumb]]
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form the sphere whose centers are along the axis chosen for consideration. If the surface of the
<math display="block"> x^2 + y^2 + z^2 = R^2,</math>
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<math display="block">r(z)^2 = x^2 + y^2 = R^2 - z^2.</math>
Therefore, the moment of inertia of the
<math display="block">\begin{align}
I_{C, \text{
&= \int_{-R}^R \
&= \
&= \pi \rho\left(1 - \
&= \
\end{align}</math>
where <math display="inline">m = \frac{4}{3}\pi R^3 \rho</math> is the mass of the sphere.
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=== Rigid body ===
[[File:RollingVsInertia.gif|400px|thumb|right|The cylinders with higher moment of inertia roll down a slope with a smaller acceleration, as more of their potential energy needs to be converted into the rotational kinetic energy.]]
If a [[mechanical system]] is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis <math>\mathbf{\hat{k}}</math>
If a system of <math>n</math> particles, <math>P_i, i = 1, \dots, n</math>, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point <math>\mathbf{R}</math>, and absolute velocities <math>\mathbf{v}_i</math>:
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<math display="block">I_\mathbf{C} = \sum_{i} m_i\,\Delta r_i^2,</math>
then the equation for angular momentum simplifies to<ref name="Beer"/>{{rp|p=1028}}
<math display="block">\mathbf{L} = I_\mathbf{C} \omega \mathbf{\hat{k}}.</math>
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\end{align}</math>
Let the reference point be the center of mass <math>\mathbf{C}</math> of the system so the second term becomes zero, and introduce the moment of inertia <math>I_\mathbf{C}</math> so the kinetic energy is given by<ref name="Beer"/>{{rp|p=1084}}
<math display="block">E_\text{K} = \frac{1}{2} I_\mathbf{C} \omega^2 + \frac{1}{2} M\mathbf{V} \cdot \mathbf{V}.</math>
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where <math>\mathbf{\hat{e}}_i \times \mathbf{\hat{e}}_i = \mathbf{0}</math>, and <math>\mathbf{\hat{e}}_i \times \mathbf{\hat{t}}_i = \mathbf{\hat{k}}</math> is the unit vector perpendicular to the plane for all of the particles <math>P_i</math>.
Use the [[center of mass]] <math>\mathbf{C}</math> as the reference point and define the moment of inertia relative to the center of mass <math>I_\mathbf{C}</math>, then the equation for the resultant torque simplifies to<ref name="Beer"/>{{rp|p=1029}}
<math display="block">\boldsymbol{\tau} = I_\mathbf{C}\alpha\mathbf{\hat{k}}.</math>
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</math>
Since the center of mass is defined by
The second term in this equation is zero because <math>\mathbf{C}</math> is the center of mass. Introduce the skew-symmetric matrix <math>[\Delta\mathbf{r}_i]</math> so the kinetic energy becomes▼
<math>
\sum_{i=1}^n m_i \Delta\mathbf{r}_i =0</math>
▲
<math display="block">\begin{align}
E_\text{K}
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\boldsymbol{\tau}
&= \sum_{i=1}^n (\mathbf{r_i} - \mathbf{R})\times (m_i\mathbf{a}_i) \\
&= \sum_{i=1}^n
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
\end{align}</math>
In the last statement,
<math display="block">\begin{align}
\boldsymbol{\tau} &= \sum_{i=1}^n m_i [
\boldsymbol{\tau} &= \sum_{i=1}^n m_i [\Delta\mathbf{r}_i\times (\boldsymbol{\alpha} \times \Delta\mathbf{r}_i) + \Delta\mathbf{r}_i \times (\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \Delta\mathbf{r}_i))]
\end{align}</math>
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<math display="block">\begin{align}
0 &=
&=
&=
&=
0 &=
\end{align}</math>
The result of applying [[Jacobi identity]] can then be continued as follows:
<math display="block">\begin{align}
&= -[\boldsymbol\omega\times((\boldsymbol\omega\times
&= -[(\boldsymbol\omega\times
&= -[(\boldsymbol\omega\times
&= -[(\boldsymbol\omega\times
&= -[(\boldsymbol\omega\times
&= -[\boldsymbol\omega\times(
&= \boldsymbol\omega\times -(
&= \boldsymbol\omega\times -(
\end{align}
</math>
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<math display="block">\begin{align}
\boldsymbol\tau
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
\end{align}</math>
<math display="block">\begin{align}
\boldsymbol{\tau}
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&= \sum_{i=1}^n m_i [
&=
&= -\sum_{i=1}^n m_i [
\boldsymbol\tau &= -\sum_{i=1}^n m_i [\Delta\mathbf{r}_i\times (\Delta\mathbf{r}_i \times \boldsymbol\alpha)] + \boldsymbol\omega\times -\sum_{i=1}^n m_i [\Delta\mathbf{r}_i \times (\Delta\mathbf{r}_i \times \boldsymbol\omega)]\;\ldots\;\boldsymbol\omega\text{ is not characteristic of particle } P_i
\end{align}</math>
Notice that for any vector <math>\mathbf{u}</math>, the following holds:
<math display="block">\begin{align}
-\sum_{i=1}^n m_i [
&= -\sum_{i=1}^n m_i \left(\begin{bmatrix}
0 & -\Delta r_{3,i} & \Delta r_{2,i} \\
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\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \\
&= -\sum_{i=1}^n m_i [\Delta r_i]^2 \mathbf{u} \\[6pt]
-\sum_{i=1}^n m_i [
&= \left(-\sum_{i=1}^n m_i [\Delta r_i]^2\right) \mathbf{u}\;\ldots\;\mathbf{u}\text{ is not characteristic of } P_i
\end{align}</math>
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<math display="block">\begin{align}
\boldsymbol{\tau}
&= -\sum_{i=1}^n m_i [
&= \left(-\sum_{i=1}^n m_i [\Delta r_i]^2\right) \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left(-\sum_{i=1}^n m_i [\Delta r_i]^2\right) \boldsymbol{\omega}
\end{align}</math>
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where
* <math>i</math>, <math>j</math> is equal to 1, 2
* <math>\mathbf{r}_k = \left(x_1^{(k)}, x_2^{(k)}, x_3^{(k)}\right)</math> is the vector to the point mass <math>m_k</math> from the point about which the tensor is calculated and
* <math>\delta_{ij}</math> is the [[Kronecker delta]].
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<math display="block">I_n = \mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n},</math>
where the [[dot product]] is taken with the corresponding elements in the component tensors. A product of inertia term such as <math>I_{12}</math> is obtained by the computation
<math display="block">I_{12} = \mathbf{e}_1\cdot\mathbf{I}\cdot\mathbf{e}_2,</math>
and can be interpreted as the moment of inertia around the <math>x</math>-axis when the object rotates around the <math>y</math>-axis.
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The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
<math display="block">\mathbf{I} = \begin{bmatrix}
I_{11} & I_{12} & I_{13} \\[1.8ex]
I_{21} & I_{22} & I_{23} \\[1.8ex]
I_{31} & I_{32} & I_{33}
\end{bmatrix} = \begin{bmatrix}
I_{xx} & I_{xy} & I_{xz} \\[1.8ex]
I_{yx} & I_{yy} & I_{yz} \\[1.8ex]
I_{zx} & I_{zy} & I_{zz}
\end{bmatrix} = \begin{bmatrix}
\sum_{k=1}^{N} m_{k} \left(y_{k}^
-\sum_{k=1}^{N} m_{k} x_{k} y_{k} &
-\sum_{k=1}^{N} m_{k} x_{k} z_{k} \\[1ex]
-\sum_{k=1}^{N} m_{k} x_{k} y_{k} &
\sum_{k=1}^{N} m_{k} \left(x_{k}^
-\sum_{k=1}^{N} m_{k} y_{k} z_{k} \\[1ex]
-\sum_{k=1}^{N} m_{k} x_{k} z_{k} &
-\sum_{k=1}^{N} m_{k} y_{k} z_{k} &
\sum_{k=1}^{N} m_{k} \left(x_{k}^
\end{bmatrix}.
</math>
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I_{yz} = I_{zy} \ &\stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} m_{k} y_{k} z_{k}, \\[3pt]
\mathbf{I} = \begin{bmatrix}
I_{11} & I_{12} & I_{13} \\[1.8ex]
I_{21} & I_{22} & I_{23} \\[1.8ex]
I_{31} & I_{32} & I_{33}
\end{bmatrix} &= \begin{bmatrix}
I_{xx} & -I_{xy} & -I_{xz} \\[1.8ex]
-I_{yx} & I_{yy} & -I_{yz} \\[1.8ex]
-I_{zx} & -I_{zy} & I_{zz}
\end{bmatrix} = \begin{bmatrix}
\sum_{k=1}^{N} m_{k} \left(y_{k}^{2} + z_{k}^{2}\right) & -\sum_{k=1}^{N} m_{k} x_{k} y_{k} & -\sum_{k=1}^{N} m_{k} x_{k} z_{k} \\[1ex]
-\sum_{k=1}^{N} m_{k} x_{k} y_{k} & \sum_{k=1}^{N} m_{k} \left(x_{k}^{2} + z_{k}^{2}\right) & -\sum_{k=1}^{N} m_{k} y_{k} z_{k} \\[1ex]
-\sum_{k=1}^{N} m_{k} x_{k} z_{k} & -\sum_{k=1}^{N} m_{k} y_{k} z_{k} & \sum_{k=1}^{N} m_{k} \left(x_{k}^{2} + y_{k}^{2}\right)
\end{bmatrix}.
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==== Determine inertia convention (Principal axes method) ====
If one has the inertia data <math>(I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{xz}, I_{yz})</math> without knowing which inertia convention that has been used, it can be determined if one also has the [[#Principal axes|principal axes]]. With the principal axes method, one makes inertia matrices from the following two assumptions:
# The standard inertia convention has been used <math>(I_{12} = I_{xy}, I_{13} = I_{xz}, I_{23} = I_{yz})</math>.
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<math display="block">I =
m \begin{bmatrix} n_1 & n_2 & n_3 \end{bmatrix} \begin{bmatrix}
y^2 + z^2 & -xy & -xz \\[0.5ex]
-yx & x^2 + z^2 & -yz \\[0.5ex]
-zx & -zy & x^2 + y^2
\end{bmatrix} \begin{bmatrix}
n_1 \\[0.7ex]
n_2 \\[0.7ex]
n_3
\end{bmatrix}.
Line 880 ⟶ 890:
=== Inertia tensor of rotation ===
Let <math>\mathbf{R}</math> be the [[rotation matrix#In three dimensions|matrix]] that represents a body's rotation. The
<math display="block">\mathbf{I} = \mathbf{R}\mathbf{I_0}\mathbf{R}^\textsf{T}</math>
Line 906 ⟶ 916:
The columns of the rotation matrix <math>\mathbf{Q}</math> define the directions of the principal axes of the body, and the constants <math>I_1</math>, <math>I_2</math>, and <math>I_3</math> are called the '''principal moments of inertia'''. This result was first shown by [[James Joseph Sylvester|J. J. Sylvester (1852)]], and is a form of [[Sylvester's law of inertia]].<ref name=syl852>{{cite journal |author=Sylvester, J J | title=A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares | journal=Philosophical Magazine |series=4th Series| volume=4 | issue=23 | pages=138–142 | year=1852 | url=http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf | doi= 10.1080/14786445208647087 | access-date=June 27, 2008}}</ref><ref name=norm>{{cite book| author=Norman, C.W.| title=Undergraduate algebra | publisher=[[Oxford University Press]] | pages=360–361 | year=1986 | isbn=0-19-853248-2 }}</ref> The principal axis with the highest moment of inertia is sometimes called the '''figure axis''' or '''axis of figure'''.
A toy [[Spinning top|top]] is an example of a rotating rigid body, and the word ''top'' is used in the names of types of rigid bodies. When all principal moments of inertia are distinct, the principal axes through [[center of mass]] are uniquely specified and the rigid body is called an '''asymmetric top'''. If two principal moments are the same, the rigid body is called a '''symmetric top''' and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a '''spherical top''' (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.
The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order <math>m</math>, meaning it is symmetrical under rotations of {{math|[[turn (geometry)|360°]]/''m''}} about the given axis, that axis is a principal axis. When <math>m > 2</math>, the rigid body is a symmetric top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other [[Platonic solid]].
Line 914 ⟶ 924:
A practical example of this mathematical phenomenon is the routine automotive task of [[Tire balance|balancing a tire]], which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.
[[Rotational spectroscopy#Classification of molecular rotors|Rotating molecules are also classified]] as asymmetric, symmetric, or spherical tops, and the structure of their [[Rotational spectroscopy|rotational spectra]] is different for each type.
===Ellipsoid===
Line 971 ⟶ 981:
== External links ==
{{Commons category|Moments of inertia}}
* [http://www.lightandmatter.com/html_books/0sn/ch04/ch04.html Angular momentum and rigid-body rotation in two and three dimensions]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html Lecture notes on rigid-body rotation and moments of inertia]
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* [http://hypertextbook.com/physics/mechanics/rotational-inertia/ Tutorial on finding moments of inertia, with problems and solutions on various basic shapes]
* [https://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/ Notes on mechanics of manipulation: the angular inertia tensor]
* [https://civilengineeronline.com/str/micalc.htm Calculation tools for moment of inertia of plane sections]▼
{{Classical mechanics derived SI units}}
{{Tensors}}
{{Authority control}}
▲* [https://civilengineeronline.com/str/micalc.htm
[[Category:Physical quantities]]▼
[[Category:Rigid bodies]]
[[Category:Rotation]]
[[Category:Articles containing video clips]]
[[Category:Moment (physics)]]
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