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\includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-3_586_527_1030_810} A uniform lamina \(A B C\) is in the form of a major segment of a circle with centre \(O\) and radius 0.35 m . The straight edge of the lamina is \(A B\), and angle \(A O B = \frac { 2 } { 3 } \pi\) radians (see diagram).

1. Show that the centre of mass of the lamina is 0.0600 m from \(O\), correct to 3 significant figures. The weight of the lamina is 14 N . It is placed on a rough horizontal surface with \(A\) vertically above \(B\) and the lowest point of the arc \(B C\) in contact with the surface. The lamina is held in equilibrium in a vertical plane by a force of magnitude \(F \mathrm {~N}\) acting at \(A\).
2. Find \(F\) in each of the following cases:
   (a) the force of magnitude \(F \mathrm {~N}\) acts along \(A B\);
   (b) the force of magnitude \(F \mathrm {~N}\) acts along the tangent to the circular arc at \(A\).