13.2: The Moment of Inertia Tensor
- Page ID
- 18090
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The angle \(\theta\) increases in time (if you push hard enough1) in accordance with
\[\vec{T}=\underset{\sim}{I}\vec{\alpha},\label{eqn:1} \]
in which \(\vec{\alpha}\) is the angular acceleration and \(\underset{\sim}{I}\) is a matrix called the moment of inertia. For the simple case shown in Figure 13.1.1, \(\underset{\sim}{I}\) is proportional to the identity matrix \(\underset{\sim}{\delta}\), \(\vec{\alpha}\) is parallel to the axis of rotation (the bolt), and its magnitude \(|\vec{\alpha}|\) is \(d^2\theta/dt^2\).
The general definition of the moment of inertia matrix is
\[I_{i j}=\int_{V} d V \rho(\vec{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right),\label{eqn:2} \]
where \(\rho(\vec{x})\) is the density (mass per unit volume). Details can be found in most classical mechanics texts, e.g., Marion (2013).
The particular case illustrated in Figure \(\PageIndex{1}\) is the rotation of a rectangular prism, with uniform density and edge dimensions \(a\), \(b\) and \(c\), about the \(\hat{e}^{(1)}\) axis. In this case both torque and angular acceleration are parallel to \(\hat{e}^{(1)}\), and the only nonzero component of \(\underset{\sim}{I}\) is \(I_{11}\), computed as follows:
\[\begin{aligned}
I_{11} &=\int_{V} d V \rho\left(x_{2}^{2}+x_{3}^{2}\right) \\
&=\rho \int_{-a / 2}^{a / 2} d x_{1} \int_{-b / 2}^{b / 2} d x_{2} \int_{-c / 2}^{c / 2} d x_{3}\left(x_{2}^{2}+x_{3}^{2}\right) \\
&=\rho \frac{a b c\left(b^{2}+c^{2}\right)}{12}.
\end{aligned} \nonumber \]
For the simple case of a cube with \(a\) = \(b\) = \(c\) = \(\Delta\),
Is the moment of inertia matrix \(\underset{\sim}{I}\) a tensor? We would expect so, since it connects two physically real vectors via Equation \(\ref{eqn:1}\). We can also establish this directly from Equation \(\ref{eqn:2}\), the general formula for \(\underset{\sim}{I}\). Like any other integral, \(\underset{\sim}{I}\) can be written as the limit of a sum:
\[I_{i j}=\sum \Delta V \rho\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right), \nonumber \]
where each term in the sum is evaluated at the center of a volume element \(\Delta V\). Now \(\Delta V\) and \(\rho\) are scalars, and so is the dot product \(x_k x_k\) (section 3.2). Moreover, we know that both \(\delta_{ij}\) and the dyad \(x_ix_j\) transform according to Equation 3.3.8. Each term in the sum is therefore a tensor, and so then is the sum itself. Taking the limit as \(\Delta V\rightarrow 0\), we conclude that \(\underset{\sim}{I}\) transforms according to Equation 3.3.8. We therefore refer to \(\underset{\sim}{I}\) as the moment of inertia tensor.
1In the case shown here, \(\vec{F}\) is really the sum of the force exerted by the person and the opposing force exerted by friction, and similarly for \(\vec{T}\).