A cockroach crawls with constant speed in a circular path of radius b on a phonograph turntable rotating with constant angular speed w.The circular path is concentric with the center of the turntable.If the mass of the insect is m and the coefficient of static friction with the surface of the turntable is (us),how fast ,relative to the turntable,can the cockroach crawl before of starts to slip if it goes (a) in the direction of the rotation
(b) in the opposite to the direction of rotation.

As angular velocity of the bug in the absolute frame of reference increases, "centripetal" force becomes larger and eventually larger than static friction. The bug starts slipping when F(friction)=F(centripetal). Kmg=m(W^2)b, here W is angular velocity of the bug in the absolute reference frame, i.e. not relative to the surface angular velocity. If the bug does not crawl, then the maximum W at which it does not slip is: W=sqrt(kg/b), k - friction coefficient. The absolute W = w - (or +) v/b, depending on the relative directions.