7 A particle \(P\) of mass 0.5 kg is attached to one end of each of two identical light elastic strings of natural length 1.6 m and modulus of elasticity 19.6 N . The other ends of the strings are attached to fixed points \(A\) and \(B\) on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4.8 m and \(A\) is higher than \(B\).

1. Find the distance \(A P\) for which \(P\) is in equilibrium on the line \(A B\).
   \(P\) is released from rest at a point on \(A B\) where both strings are taut. The strings remain taut during the subsequent motion of \(P\) and \(t\) seconds after release the distance \(A P\) is \(( 2.5 + x ) \mathrm { m }\).
2. Use Newton's second law to obtain an equation of the form \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = k x\). State the property of the constant \(k\) for which the equation indicates that \(P\) 's motion is simple harmonic, and find the period of this motion.
3. Given that \(x = 0.5\) when \(t = 0\), find the values of \(x\) for which the speed of \(P\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). {www.ocr.org.uk}) after the live examination series.
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