A rough horizontal plate rotates with a constant angular velocity `w` about its vertical axis. A particle of mass `m` lies on the plate at a distance `(5a)/(4)` from the axes. If the coefficient of friction is `(1)/(3)` and the particle remains at rest relative to the plane, show that `w le sqrt((4g)/(15a))`. The particle is now connected to axis by a horizontal elastic string, of natural length a and modulus 3 mg. If the particle remains at rest relative to the plate and at a distance `(5a)/(4)`, show that the greatest possible angular velocity is `sqrt((13)/(15).(g)/(a))` and find the least possible velocity.