Area moment (polar) of a circular lamina Find the second moment of area of a thin circular lamina of diameter d about an axis perpendicular to its plane and passing through its centre.

Introduction / Context:
The polar second moment of area of a circle about its centre is used in torsion (for circular shafts) and in combined bending-torsion problems. For a thin lamina, it is purely a geometric property of the area.

Given Data / Assumptions:

• Circular lamina, diameter d (radius r = d/2).
• Axis: through the centre, perpendicular to the plane (polar axis).
• Thin, homogeneous area (area property only).

Concept / Approach:
For a circle of radius r, the polar second moment J about the centre is J = I_x + I_y, where I_x = I_y = (π r^4) / 4. Hence J = (π r^4) / 2. Substitute r = d/2 to express in diameter form.

Step-by-Step Solution:

Verification / Alternative check:
Rectangular components: I_x = I_y = (π r^4) / 4. Adding gives (π r^4)/2, consistent with the polar formula.

Why Other Options Are Wrong:

• (π d^4)/64: This equals I_x or I_y expressed in d, not the polar J.
• (π d^4)/16, (π d^4)/8, (π d^4)/4: Overestimate J; incorrect factors.

Common Pitfalls:
Confusing polar J with planar I_x or I_y; forgetting that J = I_x + I_y for axes through the same point.

Final Answer:
J = (π d^4) / 32