1. A light inextensible string has length \(8 a\). One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to a fixed point \(B\), with \(A\) vertically above \(B\) and \(A B = 4 a\). A small ball of mass \(m\) is attached to a point \(P\) on the string, where \(A P = 5 a\).

The ball moves in a horizontal circle with constant speed \(v\), with both \(A P\) and \(B P\) taut.
The string will break if the tension in it exceeds \(\frac { 3 m g } { 2 }\)
By modelling the ball as a particle and assuming the string does not break,

1. show that \(\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }\)
2. find the least possible time needed for the ball to make one complete revolution.