Using a rectangular coordinate system, which theorem relates area moments about parallel axes?

Using a rectangular coordinate system, which theorem relates area moments about parallel axes?

• A. Area Moment Of Inertia
• B. Parallel Axis Theorem
• C. Radius Of Gyration
• D. None Of The Above

Answer: (B)

The idea being tested is how moments of area change when the axis is moved to a parallel location. If you know the moment of inertia about an axis through the centroid, you can find it about any other axis that is parallel to that centroidal axis by adding Ad^2, where A is the area of the region and d is the distance between the axes. In formula form: I_parallel = I_centroid + A d^2. This works because shifting the axis adds the same extra squared-distance contribution to every area element, and the cross terms cancel due to symmetry about the centroid. So the centroidal moment plus the extra term accounts for the whole shift. The other terms in the options describe different ideas: the area moment of inertia is the quantity itself, not the rule for shifting axes; the radius of gyration is a derived quantity (sqrt(I/A)) and doesn’t define how I changes with axis location; and “none of the above” isn’t needed because the parallel axis relation is exactly the stated rule.

The idea being tested is how moments of area change when the axis is moved to a parallel location. If you know the moment of inertia about an axis through the centroid, you can find it about any other axis that is parallel to that centroidal axis by adding Ad^2, where A is the area of the region and d is the distance between the axes. In formula form: I_parallel = I_centroid + A d^2.

This works because shifting the axis adds the same extra squared-distance contribution to every area element, and the cross terms cancel due to symmetry about the centroid. So the centroidal moment plus the extra term accounts for the whole shift.

The other terms in the options describe different ideas: the area moment of inertia is the quantity itself, not the rule for shifting axes; the radius of gyration is a derived quantity (sqrt(I/A)) and doesn’t define how I changes with axis location; and “none of the above” isn’t needed because the parallel axis relation is exactly the stated rule.