Centroid of a solid right circular cone: Along its axis of symmetry, at what height above the base does the centre of gravity (C.G.) of a solid cone of height h lie?

Introduction / Context:

Locating the centre of gravity (centroid) of common 3D bodies is a staple in engineering mechanics because it directly affects support reactions, stability, and dynamic response. A right circular cone is a standard test case that distinguishes between solid bodies and thin shells.

Given Data / Assumptions:

• Solid right circular cone of height h and uniform density.
• We measure the centroid position along the axis of symmetry from the base plane.
• No cavities or truncations; this is a full cone, not a frustum.

Concept / Approach:

For a solid cone, the centroid lies at a quarter of the height from the base along the axis. This result can be derived by integrating contributions of thin circular disks stacked along the height, whose areas scale with the square of radius (and hence linearly with height for a right cone).

Step-by-Step Solution:

Verification / Alternative check:

Contrast with a thin hollow cone (conical shell): its centroid lies at h/3 above the base, which is higher because mass is concentrated near the surface rather than the interior. This cross-check prevents mixing formulas.

Why Other Options Are Wrong:

• h/2 places the centroid at mid-height—incorrect for a cone narrowing toward the apex.
• h/3 is for a hollow (thin) conical shell, not a solid cone.
• h/5 and h/6 are too close to the base and do not match the derived result.

Common Pitfalls:

• Confusing solid vs. hollow cone centroid locations (h/4 vs. h/3).
• Measuring from the wrong reference (apex vs. base).

Final Answer:

h/4