11 In this question use \(g\) as \(10 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} A smooth plane is inclined at \(30 ^ { \circ }\) to the horizontal.
The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-18_417_709_774_669} The natural length of the string \(A P\) is 1.4 metres.
When the extension of the string \(A P\) is \(e _ { A }\) metres, the tension in the string \(A P\) is \(7 e _ { A }\) newtons.
The natural length of the string \(B P\) is 1 metre.
When the extension of the string \(B P\) is \(e _ { B }\) metres, the tension in the string \(B P\) is \(9 e _ { B }\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position.
The particle \(P\) is then released from rest.
At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres. 11

1. Show that during the subsequent motion the object satisfies the equation $$\ddot { x } + 50 x = 0$$ Fully justify your answer. 11
2. The experiment is repeated in a large tank of oil.
   During the motion the oil causes a resistive force of \(k v\) newtons to act on the particle, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the particle. The oil causes critical damping to occur.
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3. 1. Show that \(k = \frac { 16 \sqrt { 2 } } { 5 }\)
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4. (ii) Find \(x\) in terms of \(t\), giving your answer in exact form.
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5. (iii) Calculate the maximum speed of the particle.