13.4: Inertia Tensor
- Page ID
- 9629
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The square bracket term in \((13.3.9)\) is called the moment of inertia tensor, \(\mathbf{I}\), which is usually referred to as the inertia tensor
\[I_{ij} \equiv \sum^{N}_{\alpha} m_{\alpha} \left[ \delta_{ij} \left( \sum^3_k x^2_{\alpha , k} \right) − x_{\alpha , i} x_{\alpha , j} \right] \label{13.12}\]
In most cases it is more useful to express the components of the inertia tensor in an integral form over the mass distribution rather than a summation for \(N\) discrete bodies. That is,
\[I_{ij} = \int\rho (\mathbf{r}^{\prime} ) \left( \delta_{ij} \left( \sum^3_k x^2_{k} \right) − x_{i} x_{ j} \right) dV\]
The inertia tensor is easier to understand when written in cartesian coordinates \(\mathbf{r}^{\prime}_{\alpha} = (x_{\alpha}, y_{\alpha}, z_{\alpha})\) rather than in the form \(\mathbf{r}^{\prime}_{\alpha} = (x_{\alpha ,1}, x_{\alpha ,2}, x_{\alpha ,3})\). Then, the diagonal moments of inertia of the inertia tensor are
\[\begin{align}
I_{x x} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-x_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[y_{\alpha}^{2}+z_{\alpha}^{2}\right] \\[4pt] \notag
I_{y y} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-y_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+z_{\alpha}^{2}\right] \\[4pt]
I_{z z} & \equiv \sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}+z_{\alpha}^{2}-z_{\alpha}^{2}\right]=\sum_{\alpha}^{N} m_{\alpha}\left[x_{\alpha}^{2}+y_{\alpha}^{2}\right]
\notag \end{align}\]
while the off-diagonal products of inertia are
\[\begin{align} I_{yx} & = I_{xy} \equiv - \sum^N_{\alpha} m_{\alpha} [x_{\alpha} y_{\alpha}] \\[4pt] \notag I_{zx} & = I_{xz} \equiv - \sum^N_{\alpha} m_{\alpha} [x_{\alpha} z_{\alpha}] \\[4pt] \notag I_{zy} & = I_{yz} \equiv - \sum^N_{\alpha} m_{\alpha} [y_{\alpha} z_{\alpha}] \end{align}\]
Note that the products of inertia are symmetric in that
\[I_{ij} = I_{ji}\]
The above notation for the inertia tensor allows the angular momentum \ref{13.12} to be written as
\[L_i = \sum^3_j I_{ij} \omega_j \]
Expanded in cartesian coordinates
\[\begin{align} L_x & = I_{xx} \omega_x + I_{xy} \omega_y + I_{xz} \omega_z \\[4pt] \notag L_y & = I_{yx} \omega_x + I_{yy} \omega_y + I_{yz} \omega_z \\[4pt] \notag L_z & = I_{zx} \omega_x + I_{zy} \omega_y + I_{zz} \omega_z \end{align}\]
Note that every fixed point in a body has a specific inertia tensor. The components of the inertia tensor at a specified point depend on the orientation of the coordinate frame whose origin is located at the specified fixed point. For example, the inertia tensor for a cube is very different when the fixed point is at the center of mass compared with when the fixed point is at a corner of the cube.