Centroid of a right circular solid cone: For a solid cone of height h, measured along the central vertical axis from its base (the circular face), the centre of gravity (centroid) lies at what distance from the base?

Introduction / Context:
Centroid locations are essential in calculating moments, hydrostatic forces, and stresses. For a homogeneous right circular solid cone, the centroid lies along the axis of symmetry between the base and the apex at a fixed ratio of the height.

Given Data / Assumptions:

• Uniform density, solid cone.
• Height = h; base is the circular face.
• Distance measured upward from the base along the axis.

Concept / Approach:
By standard centroid formulas or integration (using similar discs), the y-coordinate of the centroid from the base is one-quarter of the height for a solid cone. This differs from a conical frustum or a hollow (conical shell), which have different centroid positions.

Step-by-Step Solution (outline):
Consider the cone as stacked thin disks of radius proportional to their distance from the apex.Compute ȳ = (∫ y dA or y dV) / (∫ dA or dV) using volume elements.Evaluation yields ȳ = h/4 from the base (or 3h/4 from the apex).

Verification / Alternative check:
Rule-of-thumb: for simple solids of revolution, the cone centroid is at 1/4 from the base, cylinder at 1/2, hemisphere (solid) at 3/8 from the base. The cone result is consistent with these benchmarks.

Why Other Options Are Wrong:

• h/2: Mid-height; correct for uniform prism/cylinder, not a cone.
• h/3: Applies to triangular area centroid, not the solid cone volume.
• h/6: Too close to the base; not supported by integration.

Common Pitfalls:

• Confusing area centroid of a triangular cross-section with volume centroid of the cone.
• Measuring from the apex instead of the base; from the apex the distance is 3h/4.

Final Answer:
h/4