Area properties: Polar moment of inertia of a circular section about an axis perpendicular to the plane through its centroid. State the correct expression in terms of radius r or diameter d. Choose the correct option.
Mechanical Engineering
Engineering Mechanics
Difficulty: Easy
Choose an option
Answer
Correct Answer: J = (π r^4) / 2
Explanation
Given/Definitions
- For a circular area of radius r, the centroidal second moments are Ix = Iy = (π r4)/4.
- The polar moment about the perpendicular centroidal axis is J = Ix + Iy.
Step-by-step calculation J = Ix + Iy = (π r4)/4 + (π r4)/4 = (π r4)/2. In terms of diameter d (r = d/2): J = (π (d/2)4)/2 = (π d4)/32.
Verification Option (a) matches the r–form; the equivalent d–form is option (d).
Final Answer J = (π r^4) / 2 (equivalently, (π d^4)/32).