1 One end of a light inextensible string of length 0.8 m is attached to a particle \(P\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3 \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circular path with centre \(O\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{894be707-4f7b-4647-b209-805522556196-2_686_586_450_248} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downward vertical through \(O\).

1. Show that at this instant the speed of \(P\) is \(4.5 \mathrm {~ms} ^ { - 1 }\).
2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant.
3. Find the magnitude of the tangential acceleration of \(P\) at this instant.