Centre of mass by integration — A/L Combined Mathematics

For continuous bodies, the centre of mass is obtained by integrating element mass moments over the body: x̄ = (∫ x dm) / (∫ dm). Standard results established in the A/L curriculum include: (1) Thin uniform circular wire arc subtending angle 2α at centre: distance from centre to G is x̄ = (r sin α) / α (for a semicircle α = π/2, giving 2r/π); (2) Uniform circular sector lamina subtending 2α: x̄ = (2r sin α) / (3α) (for a semicircular lamina, 4r / (3π)); (3) Uniform solid hemisphere of radius r: x̄ = 3r/8 from centre along the axis of symmetry; (4) Uniform hollow hemispherical shell: x̄ = r/2; (5) Uniform solid right circular cone of height h: x̄ = h/4 from base along axis; (6) Uniform hollow right circular cone: x̄ = h/3 from base.

Check before you answer (A/L Applied Maths Paper 2)

1. Mass Element Definition: Expressing dm correctly in terms of density (linear λ, surface σ, or volume ρ) and differential coordinates (e.g., dm = ρ π y² dx).
2. Integration Limits Execution: Setting correct definite integration limits from 0 to h or 0 to r and executing standard polynomial integrals.
3. Memorized Standard Results: Directly citing and applying the standard centre of mass formulas for hemispheres, cones, and sectors in composite problem questions.