6. A small ball is attached to one end of a spring. The ball is modelled as a particle of mass 0.1 kg and the spring is modelled as a light elastic spring \(A B\), of natural length 0.5 m and modulus of elasticity 2.45 N . The particle is attached to the end \(B\) of the spring. Initially, at time \(t = 0\), the end \(A\) is held at rest and the particle hangs at rest in equilibrium below \(A\) at the point \(E\). The end \(A\) then begins to move along the line of the spring in such a way that, at time \(t\) seconds, \(t \leq 1\), the downward displacement of \(A\) from its initial position is 2 sin \(2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of the particle below \(E\) is \(y\) metres.

1. Show, by referring to a simple diagram, that \(y + 0.2 = x + 2 \sin 2 t\).
2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 49 y = 98 \sin 2 t\). Given that \(y = \frac { 98 } { 45 } \sin 2 t\) is a particular integral of this differential equation,
3. find \(y\) in terms of \(t\).
4. Find the time at which the particle first comes to instantaneous rest.