A uniform cube of mass $m$ and side $a$ rests on a rough horizontal floor. A horizontal force F is applied at the middle of one top edge, at right angles to that edge, as shown in the figure. Assuming the floor is rough enough that the cube does not slide, the smallest F that starts to tip the cube over its far bottom edge, and the smallest coefficient of friction that allows tipping without sliding, are

Fig. 1: A cube of side a on a rough floor, with a horizontal force F at its top edge and its weight mg at the centre. The far bottom edge is marked as the tipping edge.