Edexcel A-Level Mathematics A2 M3.5 Statics of Rigid Bodies Questions

Practise rigid body statics through centres of mass, composite shapes, suspension angles and toppling on rough inclined planes.

Exam points

find centres of mass using integration, symmetry or composite-area moments
use the centre of mass line to calculate suspension or toppling angles

Covered syllabus points

M3.5.1 - Centre of mass of uniform rigid M3.5.2 - Simple cases of equilibrium of rigid

[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}

[Maximum number: 8]

Figure 3

A thin uniform right hollow cylinder, of radius 2 a and height k a, has a base but no top. A thin uniform hemispherical shell, also of radius 2 a, is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container C. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of C is O, as shown in Figure 3.

(a)

Show that the distance from O to the centre of mass of C is

\frac{\left(k^{2}+4 k+4\right)}{2(k+3)} a

The container is placed with its circular base on a plane which is inclined at $30^{\circ}$ to the horizontal. The plane is sufficiently rough to prevent C from sliding. The container is on the point of toppling.

[ 5 ]

Part	cylinder base	cylindrical side	hemispherical shell	C
mass ratio	4πa²	4πa·ka	8πa²	12πa²+4kπa²

(b)

Find the value of k.

[ 3 ]

$\tan60^\circ=\frac{\frac{k^2+4k+4}{2(k+3)}a}{2a}$ $k^2+4k(1-\sqrt3)+(4-12\sqrt3)=0$
Marking guidance for this part:
M1 for using $\tan60^\circ=\frac{\bar y}{2a}$ , or the equivalent $\tan30^\circ$ relationship.
A1ft for substituting the centre of mass expression from part (a).
A1 for obtaining the correct three-term quadratic.

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)