Edexcel IAL Mathematics M2.2.2 centre of mass of uniform plane laminas

Practise uniform plane laminas by splitting templates into rectangles, triangles or sectors, then taking area moments to locate the centre of mass.

Exam points

Decompose the lamina into standard areas, then take moments about an edge to find d or x-coordinate.
Use the centre-of-mass position to solve hanging or pivot conditions for angles, forces or ranges.

[Maximum number: 5]

Show that $d=\frac{4 a}{\pi}$

Masses proportional to areas: $8\pi a^2, 2\pi a^2, 2\pi a^2$

Distances from AC: $\frac{16a}{3\pi}, \frac{8a}{3\pi}, 8a-\frac{8a}{3\pi}$

8\pi a^2\left(\frac{16a}{3\pi}\right)-2\pi a^2\left(\frac{8a}{3\pi}\right)-2\pi a^2\left(8a-\frac{8a}{3\pi}\right)=8\pi a^2 d

\frac{32a^3}{\pi}=8a^2 d

A1*

d=\frac{4a}{\pi}

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