Question A.05: For the circular area of Fig. A.18, determine (a) the polar ...
For the circular area of Fig. A.18, determine (a) the polar moment of inertia J_{O},(b) the rectangular moments of inertia I_{x} and I_{y}.
(a) Polar Moment of Inertia. We select as an element of area a ring of radius \rho and thickness d \rho (Fig. A.19). Since all the points within the ring are at the same distance \rho from the origin O, the polar moment of inertia of the ring is
d J_{O}=\rho^{2} d A=\rho^{2}(2 \pi \rho d \rho)
Integrating in \rho from 0 to c, we write
\begin{aligned} & J_{O}=\int_{A} \rho^{2} d A=\int_{0}^{c} \rho^{2}(2 \pi \rho d \rho)=2 \pi \int_{0}^{c} \rho^{3} d \rho \\ & J_{O}=\frac{1}{2} \pi c^{4} \end{aligned}
(b) Rectangular Moments of Inertia. Because of the symmetry of the circular area, we have I_{x}=I_{y}. Recalling Eq. (A.9), we write
J_{O}=I_{x}+I_{y}=2 I_{x} \quad \frac{1}{2} \pi c^{4}=2 I_{x}
and, thus,
I_{x}=I_{y}=\frac{1}{4} \pi c^{4}
