Centroid of a thin hollow cone: For a thin conical surface of height h, the center of gravity measured along the axis above the base lies at what distance?

Introduction / Context:

Locating centroids is essential for computing moments, stability, and support reactions. A thin hollow cone (a conical shell) has mass concentrated on its surface, not its volume, so its centroid differs from that of a solid cone.

Given Data / Assumptions:

• Geometric shape: thin conical surface of height h and negligible thickness.
• Uniform surface density.
• Axis of symmetry is vertical; base is a circle.

Concept / Approach:

For a solid right circular cone, the centroid lies at h/4 from the base along the axis. For a thin conical shell, the centroid shifts upward because the mass is distributed near the surface farther from the axis compared to volume distribution. The standard result for a thin hollow cone is h/3 above the base along the axis.

Step-by-Step Solution:

Verification / Alternative check:

The value h/3 lies above h/4 (solid cone), consistent with intuition that a shell’s mass is farther from the base centroid than a solid’s mass average.

Why Other Options Are Wrong:

• h/4 is for a solid cone, not a shell.
• h/2, 2h/3, 3h/4 place the centroid unrealistically high for this geometry.

Common Pitfalls:

• Confusing solid and hollow cone centroid formulas.

Final Answer:

h / 3