Centre of mass by integration — A/L Combined Mathematics

What must I remember about Centre of mass by integration?

• Continuous Mass Center via Calculus: xG = (1/M) ∫ x dm, where M = ∫ dm is total mass and dm is an infinitesimal mass element.
• Uniform Circular Arc (angle 2α, radius a): xG = a * (sin α) / α from center along line of symmetry; for semi-circular wire (α = π/2): xG = 2a / π.
• Uniform Circular Sector (angle 2α, radius a): xG = (2/3)a * (sin α) / α from center along line of symmetry; for semi-circular lamina (α = π/2): xG = 4a / (3π).
• Uniform Hemispherical Shell vs Solid Hemisphere: Hollow hemispherical shell of radius a: xG = a / 2 from center; Solid uniform hemisphere of radius a: xG = 3a / 8 from center.
• Uniform Solid Cone vs Hollow Cone: Hollow conical shell of height h: xG = h / 3 from base along axis; Solid uniform right circular cone of height h: xG = h / 4 from base along axis.

Test yourself (1 minute)

1. Solid Hemisphere derivation setup

   Slice solid hemisphere into thin circular discs of thickness dx at distance x from center; radius of disc is y = √(a² - x²); volume element dV = π y² dx = π(a² - x²)dx; xG = ∫₀^a x π(a² - x²)dx / (2/3 π a³) = (π a⁴ / 4) / (2/3 π a³) = 3a / 8.

2. Hollow vs Solid C.M. rule

   For corresponding geometrical shapes, the C.M. of a hollow shell is always further from the apex/center (closer to mass concentration) than the solid counterpart (e.g., solid cone h/4 vs hollow cone h/3).

3. Angle in radians requirement

   In formulas a(sin α)/α and (2/3)a(sin α)/α, the semi-vertical angle α MUST be expressed in RADIANS.

Practise Centre of mass by integration questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 5

What should I study first?

What comes next?

Next in the syllabus

Where does Centre of mass by integration fit in the syllabus?

Year 2 · Term 2