Centre of gravity of a thin hollow cone (conical surface) For a thin hollow right circular cone of total height H, the centre of gravity lies along the axis of symmetry at what height above the base?

Introduction / Context:
Locating the centre of gravity (CG) of conical shells is important in vessel design, antenna reflectors, and sheet-metal cones. The thin hollow cone refers to a conical surface (not a solid), changing the CG location compared to a solid cone.

Given Data / Assumptions:

• Uniform thin conical surface (negligible thickness).
• Total height H measured from base plane to apex.
• Axisymmetric geometry about the cone axis.

Concept / Approach:

For a thin conical surface, the area element distribution leads to the classical result: the CG lies at one-third of the height above the base along the axis (measured from the base plane). This differs from a solid cone, whose CG is at H/4 from the base.

Step-by-Step Solution:

Verification / Alternative check:

Reference centroid tables: Thin conical surface → z̄ = H/3 from base; Solid cone → z̄ = H/4 from base. Distinguishing these two is a common exam checkpoint.

Why Other Options Are Wrong:

(a) H/2 is too high, corresponding to neither solid nor surface cone. (c) H/4 belongs to a solid cone, not a thin shell. (d) is invalid because a definitive location exists. (e) 2H/3 is measured from the apex direction if misinterpreted; from the base it is incorrect for a hollow cone.

Common Pitfalls:

Mixing up solid versus hollow cone results; measuring from the apex instead of the base.

Final Answer:

one-third of the total height above base