6 A horizontal disc with a rough surface rotates about a fixed vertical axis which passes through the centre of the disc. A particle \(P\) of mass 0.2 kg is in contact with the surface and rotates with the disc, without slipping, at a distance 0.5 m from the axis. The greatest speed of \(P\) for which this motion is possible is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the coefficient of friction between the disc and \(P\).
   \(P\) is now attached to one end of a light elastic string, which is connected at its other end to a point on the vertical axis above the disc. The tension in the string is equal to half the weight of \(P\). The disc rotates with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) rotates with the disc without slipping. \(P\) moves in a circle of radius 0.5 m , and the taut string makes an angle of \(30 ^ { \circ }\) with the horizontal.
2. Find the greatest and least values of \(\omega\) for which this motion is possible.
3. Calculate the value of \(\omega\) for which the disc exerts no frictional force on \(P\).