A particle \(P\) of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N . The ends of the string are attached to the points \(A\) and \(B\) on a smooth horizontal table, where \(A B = 2 \mathrm {~m}\). The mid-point of \(A B\) is \(O\) and the point \(C\) is on the table between \(O\) and \(B\) where \(O C = 0.2 \mathrm {~m}\). At time \(t = 0\) the particle is released from rest at \(C\). At time \(t\) seconds the length of the string \(A P\) is \(( 1 + x ) \mathrm { m }\).
Show that the tension in \(B P\) is \(\frac { 2 } { 7 } ( 3 - 10 x ) \mathrm { N }\).
Find, in terms of \(x\), the tension in \(A P\).
Show that \(P\) performs simple harmonic motion with period \(2 \pi \sqrt { } \left( \frac { 7 } { 80 } \right)\) s.
Find the greatest speed of \(P\) during the motion.
The point \(D\) lies between \(O\) and \(A\), where \(O D = 0.1 \mathrm {~m}\).
Find the time taken by \(P\) to move directly from \(C\) to \(D\).