7
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_282_474_1809_794} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.5 m below \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Show that, while the string is taut, the tension, \(T \mathrm {~N}\), in the string is given by $$T = 5.88 \cos \theta + 0.4 u ^ { 2 } - 3.92 .$$
2. Find the least value of \(u\) for which the particle will move in a complete circle.
3. If in fact \(u = 3.5 \mathrm {~ms} ^ { - 1 }\), find the speed of the particle at the point where the string first becomes slack.