7 A simple pendulum consists of a light inextensible string of length 0.8 m and a particle \(P\) of mass \(m \mathrm {~kg}\). The pendulum is hanging vertically at rest from a fixed point \(O\) when \(P\) is given a horizontal velocity of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, in the subsequent motion, the maximum angle between the string and the downward vertical is 0.107 radians, correct to 3 significant figures.
2. Show that the motion may be modelled as simple harmonic motion, and find the period of this motion.
3. Find the time after the start of the motion when the velocity of the particle is first \(- 0.2 \mathrm {~ms} ^ { - 1 }\) and find the angular displacement of \(O P\) from the downward vertical at this time.