2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length \(a\) metres, and is moving in a vertical circle with centre O and radius \(a\) metres. When P is vertically below O , its speed is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When OP makes an angle \(\theta\) with the downward vertical, and the string is still taut, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the tension in the string is \(T \mathrm {~N}\), as shown in Fig. 2.
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\caption{Fig. 2}
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- Find an expression for \(v ^ { 2 }\) in terms of \(a\) and \(\theta\), and show that
$$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
- Given that \(a = 0.9\), show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
- Find the largest value of \(a\) for which P moves in a complete circle.
- Given that \(a = 1.6\), find the speed of P at the instant when the string first becomes slack.