Edexcel IAL Mathematics M3.5.1 centres of mass of uniform rigid bodies

Practise centres of mass for rigid bodies by using integration, symmetry and composite-area or volume moments before applying the result.

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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M3.5.1 - Centre of mass of uniform rigid question 1