A uniform rod is placed leaning against a smooth vertical wall and resting on a rough horizontal floor. The coefficient of static friction between the rod and the floor is 1 2 3 . What is the minimum angle the rod can make with the horizontal floor without slipping?

Step-by-step solution

Let the mass of the rod be m , its length be L , and the angle it makes with the horizontal floor be . The forces acting on the rod are: 1. Weight mg acting vertically downwards from the center of mass ( L/2 from either end). 2. Normal reaction from the floor N_f acting vertically upwards. 3. Normal reaction from the wall N_w acting horizontally outwards. 4. Static friction f acting horizontally inwards at the floor. For translational equilibrium: F_y = 0 N_f = mg F_x = 0 f = N_w For rotational equilibrium, taking