Edexcel A-Level Mathematics A2 M3.5.1 Centre of Mass of Uniform Rigid Questions

Practise locating centres of mass for semicircles, hemispheres and solids with integrals, combining added or removed regions, and applying toppling or suspension conditions.

Exam points

Set up algebraic integrals for semicircles, hemispheres or solids to locate a centre of mass.
Combine added or removed areas and volumes before using the COM in toppling or suspension.

[Maximum number: 5]

Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius r and centre O is at a distance $\frac{4 r}{3 \pi}$ from the diameter through O [You may assume, without proof, that the area of a circle of radius r is $\pi r^{2}$ ]

A uniform lamina L is in the shape of a semicircle with centre B and diameter A C=8 a. The semicircle with diameter AB is removed from L and attached to the straight edge BC to form the template T, shown shaded in Figure 4.

Figure 4

The distance of the centre of mass of T from AC is d.

\bar{x}=\frac{\pi\int_0^r x(r^2-x^2)\,dx}{\frac{2}{3}\pi r^3}

=\frac{\pi\left[\frac{r^2x^2}{2}-\frac{x^4}{4}\right]_0^r}{\frac{2}{3}\pi r^3}

=\frac{\pi\cdot\frac{r^4}{4}}{\frac{2}{3}\pi r^3}

A1*

\bar{x}=\frac{4r}{3\pi}

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Edexcel A-Level Mathematics A2 M3.5.1 Centre of Mass of Uniform Rigid Questions question 1