13 In this question take \(\boldsymbol { g = \mathbf { 1 0 }\).} A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 m . The other end of the string is attached to a fixed point O . The particle P rests in equilibrium at a point A with the string vertical.

1. Show that the distance OA is 0.5 m . At time \(\mathrm { t } = 0 , \mathrm { P }\) is projected vertically downwards from A with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Throughout the subsequent motion, \(P\) experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) ).
2. Given that the downward displacement of P from A at time t seconds is x metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 200 x = 0$$
3. Verify that \(\mathrm { x } = \frac { 5 } { 56 } \mathrm { e } ^ { - 2 \mathrm { t } } \sin ( 14 \mathrm { t } )\).
4. Determine whether the string becomes slack during the motion.