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The diagram shows a uniform lamina \(A B C D E F\), formed from a semicircle with centre \(O\) and radius 1 m by removing a semicircular part with centre \(O\) and radius \(r \mathrm {~m}\).
- Show that the distance in metres of the centre of mass of the lamina from \(O\) is
$$\frac { 4 \left( 1 + r + r ^ { 2 } \right) } { 3 \pi ( 1 + r ) } .$$
The centre of mass of the lamina lies on the \(\operatorname { arc } A B C\).
- Show that \(r = 0.494\), correct to 3 significant figures.
The lamina is freely suspended at \(F\) and hangs in equilibrium.
- Find the angle between the diameter of the lamina and the vertical.