37.6: Lees’ Rule
- Page ID
- 92277
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Lees' rule, like Routh's rule, is a formula for computing the moment of inertia of a symmetrical solid. It is really a kind of mnemonic device for helping to recall several moment of inertia formulæ.
Lees' rule states that the moment of inertia \(I\) of a body of mass \(M\) about an axis is given by
\[I=M\left(\frac{a^{2}}{3+n}+\frac{b^{2}}{3+n^{\prime}}\right)\]
where \(a\) and \(b\) are the lengths of the semi-axes perpendicular to the rotation axis, and \(n\) and \(n^{\prime}\) are the "numbers of principal curvature" that terminate semi-axes \(a\) and \(b\), respectively ( \(n, n^{\prime}=0\) for a flat surface, 1 for a cylindrical surface, or 2 for a spherical surface).
For example, suppose we want the moment of inertia of a rectangular plate of dimensions \(\ell \times w\), about an axis through the center of the plate and perpendicular to the plane of the plate.
Solution
Then \(a=\ell / 2, b=w / 2\), and \(n=n^{\prime}=0\) because the surfaces are flat. Then Lees' rule gives
\[I=M\left(\frac{\ell^{2} / 4}{3}+\frac{w^{2} / 4}{3}\right)=\frac{1}{12} M\left(\ell^{2}+w^{2}\right)\]
As another example, consider the moment of inertia of a solid cylinder of radius \(R\) rotated about its axis.
Solution
In this case \(a=b=R\), and \(n=n^{\prime}=1\). Lees' rule in this case gives
\[I=M\left(\frac{R^{2}}{4}+\frac{R^{2}}{4}\right)=\frac{1}{2} M R^{2}\]
As a third example, consider the moment of inertia of a solid cylinder of radius \(R\) and length \(\ell\) rotated about an axis perpendicular to the cylinder axis, and passing through the center of the cylinder.
Solution
In this case, \(a=R\), \(b=\ell / 2, n=1\), and \(n^{\prime}=0\). Then Lees' rule gives
\[I=M\left(\frac{R^{2}}{4}+\frac{\ell^{2} / 4}{3}\right)=M\left(\frac{R^{2}}{4}+\frac{\ell^{2}}{12}\right)\]