Question A.5: For the circular area of Fig. A.13a, determine (a) the polar......
For the circular area of Fig. A.13a, determine (a) the polar moment of inertia J_O, (b) the rectangular moments of inertia I_x \text { and } I_y.
a. Polar Moment of Inertia. We choose to select as an element of area a ring of radius r with a thickness dr (Fig. A.13b). Since all of the points within the ring are at the same distance r 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 ρ from 0 to c
\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, I_x = I_y . Recalling Eq. (A.9), write
J_O=I_x+I_y=2 I_x \quad \frac{1}{2} \pi c^4=2 I_x
and
I_x=I_y=\frac{1}{4} \pi c^4
