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 $B$.\\
(i) 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 $( 2.5 + x ) \mathrm { m }$.\\
(ii) 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.\\
(iii) Given that $x = 0.5$ when $t = 0$, find the values of $x$ for which the speed of $P$ is $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

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\hfill \mbox{\textit{OCR M3 2010 Q7 [14]}}