Problem No 10 A thin rod of mass $m$ is placed into a corner formed by a vertical wall and a horizontal floor so that the rod forms an angle $\alpha$ with the floor and is perpendicular to the line where the floor and the wall meet. The static coefficient of friction of the rod against the wall and against the floor is $\mu=\tan\beta$, which is not large enough for keeping the current position – as long as the only forces applied to the rod are the normal and and friction forces applied to its endpoints, and the gravity force $mg$ applied to its centre. What is the minimal additional force $F$ needed for maintaining the current position of the rod (assuming that its direction and application point are optimal)? Express your answer in terms of $m, g, \alpha$, and $\beta$.