What is the polar moment of inertia J for a solid circular shaft of radius R?

• A. π R^4 / 2
• B. π R^4
• C. π R^2
• D. 2π R^4

Answer: (A)

In torsion, the cross-section’s resistance is described by the polar moment of inertia J. For a solid disk, view the disk as a stack of thin rings. A ring at radius r with thickness dr has area dA = 2π r dr, and its contribution to J is r^2 dA. Integrating over the full radius gives J = ∫0^R r^2 (2π r dr) = 2π ∫0^R r^3 dr = (π R^4)/2. This is the value that results from correctly accounting for how area grows with radius. The other options arise from incorrect handling of the area element or the r-dependence, so they don’t match the actual distribution of area in a solid circular cross-section.