Thermal Engineering, Refrigeration & Power

For a circular cross-section of diameter $d$, the rectangular area moments of inertia about the centroidal axes are $I_{xx} = I_{yy} = \frac{\pi d^4}{64}$. By the Perpendicular Axis Theorem, the polar moment of inertia ($I_p$ or $J$) about an axis perpendicular to the plane of the circle is the sum of both orthogonal moments:

Option analysis:

• $\frac{\pi d^4}{64}$: Represents planar moment of inertia ($I_{xx}$ or $I_{yy}$), not polar moment.
• $\frac{\pi d^4}{32}$: Correct polar moment of inertia of a circular section.
• $\frac{\pi d^4}{16}$: Incorrect.
• $\frac{\pi d^3}{32}$: Represents polar section modulus ($Z_p$), not moment of inertia.