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\includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-4_588_629_274_758} A particle \(P\) of mass 0.8 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . A particle \(Q\) is suspended from \(O\) by an identical string. With the string \(O P\) taut and inclined at \(\frac { 1 } { 3 } \pi\) radians to the vertical, \(P\) is projected with speed \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the string so as to strike \(Q\) directly (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 7 }\).

1. Calculate the tension in the string immediately after \(P\) is set in motion.
2. Immediately after \(P\) and \(Q\) collide they have equal speeds and are moving in opposite directions. Show that \(Q\) starts to move with speed \(0.15 \mathrm {~ms} ^ { - 1 }\).
3. Prove that before the second collision between \(P\) and \(Q , Q\) is moving with approximate simple harmonic motion.
4. Hence find the time interval between the first and second collisions of \(P\) and \(Q\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }