Composite centroid along an axis: From a solid right circular cylinder (height 8 cm, radius 4 cm) a right circular cone of the same height and base is removed (scooped) from the top. Find the height of the centre of gravity of the remaining solid measured from the base.

Introduction / Context:Finding the centre of gravity (CG) of a composite body is a standard statics task. When a cone is removed from a cylinder (same base and height), symmetry keeps the CG on the common axis; its height is determined by the principle of moments of volumes (or weights) about the base.

Given Data / Assumptions:

• Cylinder: radius r = 4 cm, height h = 8 cm; CG at h/2 = 4 cm from base.
• Removed cone: same base and height; CG at h/4 = 2 cm from base (measured along the axis from the base).
• Uniform density; use volumes as weights.

Concept / Approach:Use the composite-body formula for centroid: ȳ = (Σ V_i * y_i) / (Σ V_i), with the removed part contributing negative volume. Volumes: V_cyl = π r^2 h, V_cone = (1/3) π r^2 h.

Step-by-Step Solution:

Verification / Alternative check:Dimensional and symmetry checks: result lies between 4 cm (solid cylinder) and 6 cm (if material concentrated toward the top), consistent with removing more mass near the top than the bottom.

Why Other Options Are Wrong:

• 4.5, 5.25, 5.5 cm: Do not satisfy the exact composite centroid calculation.

Common Pitfalls:Using the wrong cone centroid location (it is at h/4 from the base, not from the apex) and forgetting to subtract the removed volume’s moment.

Final Answer:5.0 cm