A particle \(P\) of mass 0.2 kg moves away from the origin along the positive \(x\)-axis. It moves under the action of a force directed away from the origin \(O\), of magnitude \(\frac { 5 } { x + 1 } \mathrm {~N}\), where \(O P = x\) metres. Given that the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(x = 0\), find the value of \(x\), to 3 significant figures, when the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8)
One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N , is attached to a fixed point \(A\). A small ball \(B\) of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at \(A\) and first comes to instantaneous rest at the point \(C\), vertically below \(A\).
Find the distance \(A C\).
Find the instantaneous acceleration of \(B\) at \(C\).
(3)
\section*{3.}
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{a46d3d34-2381-4e73-837f-a60663fb1419-3_520_967_264_531}
A rod \(A B\), of mass \(2 m\) and length \(2 a\), is suspended from a fixed point \(C\) by two light strings \(A C\) and \(B C\). The rod rests horizontally in equilibrium with \(A C\) making an angle \(\alpha\) with the rod, where tan \(\alpha = \frac { 3 } { 4 }\), and with \(A C\) perpendicular to \(B C\), as shown in Fig. 1.
Give a reason why the rod cannot be uniform.
Show that the tension in \(B C\) is \(\frac { 8 } { 5 } m g\) and find the tension in \(A C\).
The string \(B C\) is elastic, with natural length \(a\) and modulus of elasticity \(k m g\), where \(k\) is constant.