5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point $O$ and the other end is attached to a particle $P$ of weight 1.5 N . The particle is released from rest at the point $A$, which is 2.1 m vertically below $O$. It comes instantaneously to rest at $B$, which is vertically above $O$.\\
(i) Verify that the distance $A B$ is 4 m .\\
(ii) Find the maximum speed of $P$ during its upward motion from $A$ to $B$.

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\caption{Fig. 1}
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A light inextensible string of length $0.8 \pi \mathrm {~m}$ has particles $P$ and $Q$, of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point $O$. The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with $P$ and $Q$ at opposite ends of the horizontal diameter of the cylinder through $O$ (see Fig. 1). The string is released and $Q$ begins to descend. When $O P$ has rotated through $\theta$ radians, with $P$ remaining in contact with the cylinder, the speed of each particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see Fig. 2).\\

\hfill \mbox{\textit{OCR M3 2010 Q5 [11]}}