18 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-26_439_154_685_927} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7 e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3 e\) newtons when the extension is \(e\) metres. 18

1. Find the extension of each string when the system is in equilibrium.
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2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5 v\) newtons to act on the ball, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii). 18
3. 1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 9 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 20 x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released.
      [0pt] [3 marks]
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4. (ii) Find \(x\) in terms of \(t\)
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5. State one limitation of the model used in part (b)