Non-uniform rod or wire suspended in equilibrium

2 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_686_495_1238_826} A uniform rigid wire \(A B\) is in the form of a circular arc of radius 1.5 m with centre \(O\). The angle \(A O B\) is a right angle. The wire is in equilibrium, freely suspended from the end \(A\). The chord \(A B\) makes an angle of \(\theta ^ { \circ }\) with the vertical (see diagram).

1. Show that the distance of the centre of mass of the arc from \(O\) is 1.35 m , correct to 3 significant figures.
2. Find the value of \(\theta\).

2 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_448_547_434_799} The diagram shows a uniform object \(A B C\) of weight 3 N in the form of an arc of a circle with centre \(O\) and radius 0.7 m . The angle \(A O C\) is 2 radians. The object rests in equilibrium with \(A\) on a horizontal surface and \(C\) vertically above \(A\). Equilibrium is maintained by a horizontal force of magnitude \(F \mathrm {~N}\) applied at \(C\) in the plane of the object. Calculate \(F\).

3 \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-2_542_638_1208_717} A uniform semicircular arc \(A C B\) is freely pivoted at \(A\). The arc has mass 0.3 kg and is held in equilibrium by a force of magnitude \(P\) N applied at \(B\). The line of action of this force lies in the same plane as the arc, and is perpendicular to \(A B\). The diameter \(A B\) has length 4 cm and makes an angle of \(\theta ^ { \circ }\) with the downward vertical (see diagram).

1. Given that \(\theta = 0\), find the magnitude of the force acting on the arc at \(A\).
2. Given instead that \(\theta = 30\), find the value of \(P\).