1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.

The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).

1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.