Two points \(A\) and \(B\) are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N , has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N , has one end attached to the point \(B\). A particle \(P\) of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point \(C\) on the straight line between \(A\) and \(B\).
Show that the length of \(A C\) is 1.76 m .
The particle \(P\) is now held at the point \(D\) on the line \(A B\) such that \(A D = 2.16 \mathrm {~m}\). The particle is then released from rest and in the subsequent motion both strings remain taut.
Show that \(P\) moves with simple harmonic motion.
Find the speed of \(P\) as it passes through the point \(C\).
Find the time from the instant when \(P\) is released from \(D\) until the instant when \(P\) is first moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).