On detachment conditions of a top on an absolutely rough support

The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.