Edexcel IAL Mathematics M3.5.2 equilibrium of rigid bodies

Practise rigid-body equilibrium by using centre-of-mass lines, moments and geometry for suspended templates and bodies on planes.

Exam points

Take moments about a pivot or contact edge to find an angle, force or constant.
Link the centre-of-mass line to tanθ or a toppling condition for a rigid body.

[Maximum number: 3]

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.

Find the value of k.

$\tan60^\circ=\frac{\frac{k^2+4k+4}{2(k+3)}a}{2a}$ $k^2+4k(1-\sqrt3)+(4-12\sqrt3)=0$

M1 A1 A1

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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M3.5.2 - Simple cases of equilibrium of rigid question 1