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\includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-4_529_403_264_829} A small smooth ring \(P\) of mass 0.4 kg is threaded onto a light inextensible string fixed at \(A\) and \(B\) as shown in the diagram, with \(A\) vertically above \(B\). The string is inclined to the vertical at angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) at \(A\) and \(B\) respectively. \(P\) moves in a horizontal circle of radius 0.5 m about a point \(C\) vertically below \(B\).

1. Calculate the tension in the string.
2. Calculate the speed of \(P\). The end of the string at \(B\) is moved so both ends of the string are now fixed at \(A\).
3. Show that, when the string is taut, \(A P\) is now 0.854 m correct to 3 significant figures.
   \(P\) moves in a horizontal circle with angular speed \(3.46 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
4. Find the tension in the string and the angle that the string now makes with the vertical.