10.5: Polar Moment of Inertia
- Page ID
- 70287
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- How are polar moments of inertia similar and different to area moments of inertia about either a horizontal or vertical axis?
The polar moment of inertia is defined by the integral quantity
\begin{equation} J_O = \int_A r^2 dA\text{,}\tag{10.5.1} \end{equation}
where \(r\) is the distance from the reference point to a differential element of area \(dA\text{.}\)
The polar moment of inertia describes the distribution of the area of a body with respect to a point in the plane of the body. Alternately, the point can be considered to be where a perpendicular axis crosses the plane of the body. The subscript on the symbol \(j\) indicates the point or axis.
There is a particularly simple relationship between the polar moment of inertia and the rectangular moments of inertia. Referring to the figure, apply the Pythagorean theorem \(r^2 = x^2 +y^2\) to the definition of polar moment of inertia to get
\begin{align*} J_O & = \int_A r^2\ dA\notag \\ & = \int_A (x^2 + y^2) \ dA\notag\\ &= \int_A x^2 dA + \int_A y^2 dA\notag\\ J_O & = I_x + I_y\text{.}\label{Jo}\tag{10.5.2} \end{align*}
- Thinking Deeper 10.5.1. Torsional Stress.
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The polar moment of inertia is an important factor in the design of drive shafts. When a shaft is subjected to torsion, it experiences internal distributed shearing forces throughout its cross-section which counteract the external torsional load.
This distributed shearing force is called shear stress, and is usually given the symbol tau, \(\tau\text{.}\) Shear stress is zero at the neutral axis and increases linearly with \(r\) to a maximum value, \(\tau_\text{max}\) at the outside surface where \(r=c\text{,}\) so
\[ \tau = \tau_\text{max}\frac{r}{c} \nonumber \]
Figure 10.5.2. Section-cut view of a shaft, showing shearing forces developed to withstand external torsion \(T\text{.}\)
The force at any point is \(dF = \tau\ dA\text{,}\) and the moment \(dM\) exerted at any point is \(dF\) times the moment arm, which is \(r\text{.}\) The total moment is the integral of this quantity over the area of the cross section, and is proportional to the polar moment of inertia.
\begin{align*} T &= \int dM\\ & = \int r\ dF\\ &= \int_A r\ \tau_\text{max} \frac{r}{c} \ dA\\ &= \frac{\tau_\text{max}}{c} \int_A r^2\ dA\\ T & = \frac{\tau_\text{max}}{c} J\\ \tau_\text{max} & = \frac{Tc}{J} \end{align*}
This is the relationship between the maximum stress in a circular shaft and the applied torque \(T\) and the geometric properties of the shaft, \(J_O\) and \(c\text{.}\)