Key Questions
- How are polar moments of inertia similar and different to area moments of inertia about either a horizontal or vertical axis?
Section 10.5 Polar Moment of Inertia
The polar moment of inertia is defined by the integral quantity
\begin{equation}
J_O = \int_A r^2 dA\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 \amp = \int_A r^2\ dA\notag\\
\amp = \int_A (x^2 + y^2) \ dA\notag\\
\amp = \int_A x^2 dA + \int_A y^2 dA\notag\\
J_O \amp = I_x + I_y\tag{10.5.2}
\end{align}
.
Thinking Deeper 10.5.1. Torsional Stress.
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
\begin{equation*}
\tau = \tau_\text{max}\frac{r}{c}
\end{equation*}
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 \amp= \int dM\\
\amp = \int r\ dF\\
\amp= \int_A r\ \tau_\text{max} \frac{r}{c} \ dA\\
\amp = \frac{\tau_\text{max}}{c} \int_A r^2\ dA\\
T \amp = \frac{\tau_\text{max}}{c} J\\
\tau_\text{max} \amp = \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 \(c\text{.}\)
