7 One end of a light inextensible string of length 0.8 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.8 m vertically below \(O\) with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string makes an angle \(\theta\) with the downward vertical.

1. While the string remains taut, show that \(v ^ { 2 } = 15.68 ( 1 + \cos \theta )\), and find the tension in the string in terms of \(\theta\).
2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\).
3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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