Centre of mass of uniform bodies — A/L Combined Mathematics

What must I remember about Centre of mass of uniform bodies?

• Center of Mass (C.M.) Definition: The unique point G at which the total mass M of a body or system of particles may be considered to be concentrated: M * rG = ∑ m_i * r_i.
• Cartesian coordinates of Center of Mass: xG = (∑ m_i * x_i) / M; yG = (∑ m_i * y_i) / M; zG = (∑ m_i * z_i) / M, where M = ∑ m_i.
• Symmetry Principles: 1) If a body possesses a plane of symmetry, its C.M. lies in that plane; 2) If a body possesses an axis of symmetry, its C.M. lies on that axis; 3) If a body possesses a center of symmetry, its C.M. is that center.
• Standard Uniform 1D and 2D bodies: Uniform straight rod: at midpoint (L/2); Uniform rectangular/square lamina: at intersection of diagonals; Uniform circular wire / disc: at geometric center; Uniform triangular lamina: at Centroid G (intersection of medians, 1/3 distance from base along median).
• Standard Uniform 3D bodies: Uniform solid sphere / spherical shell: at center; Uniform rectangular block / cuboid: at geometric center.

Test yourself (1 minute)

1. Mass vs Area/Length proportionality

   For uniform wires of density λ, mass m = λ * length; for uniform laminas of surface density σ, mass m = σ * area; densities cancel in xG and yG ratios.

2. Triangular Lamina vs Three-Rod Framework

   Triangular LAMINA has C.M. at centroid (h/3 from base); Triangular perimeter FRAMEWORK made of 3 rods has C.M. at INCENTER of medial triangle formed by rod midpoints.

3. Mass Moment principle

   The algebraic sum of mass moments about any plane passing through the Center of Mass is identically ZERO.

Practise Centre of mass of uniform bodies questions today

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

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Year 2 · Term 2