Revision

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

NIE competency: Finds the centre of mass composite bodies and remaining bodies

Aligned with the NIE syllabus

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

Composite Body Principle: A composite body composed of parts with masses m1, m2, ..., mn and respective centers of mass (x1, y1), (x2, y2), ..., (xn, yn) has combined C.M. at: xG = (∑ m_i * x_i) / (∑ m_i) and yG = (∑ m_i * y_i) / (∑ m_i).
Remaining Body (Subtracted Part) Principle: If a part of mass m2 with C.M. at (x2, y2) is removed from a complete body of total mass M1 with C.M. at (x1, y1), the remaining body of mass M = M1 - m2 has C.M. at: xG = (M1*x1 - m2*x2) / (M1 - m2).
Choice of Coordinate Axes: Always align axes along lines or planes of symmetry to automatically set one or more coordinates to zero (e.g., yG = 0).
Tabular format for calculations: Organize part label, mass / relative weight (w_i), coordinates of part C.M. (x_i, y_i), and mass moments (w_i * x_i, w_i * y_i) in structured columns.
Joined uniform bodies: When rods, sheets, or solids are joined, replace each component by its equivalent concentrated particle at its individual C.M. before taking overall weighted moments.

Negative mass technique for cutouts

Treat removed holes or cavity sections as objects with NEGATIVE mass m = -m2; sum normally using xG = (∑ m_i * x_i) / (∑ m_i).
Proportional mass substitution

If thickness and density are uniform, substitute Area directly for 2D laminas and Volume for 3D solids; relative ratios can be simplified by canceling common factors.
Hemi-sphere with cylinder or cone

For a toy consisting of a hemisphere of radius r joined to a cone of base radius r and height h, C.M. lies on common axis of symmetry; distance from joint interface is determined by balancing moments.

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