A uniform rod of length L rests against a smooth roller, as shown in the figure. Determine the friction coefficient between the ground and the lower end, given that the minimum angle the rod can make with the horizontal is heta .

Step-by-step solution

Let the mass of the rod be m and its length be L . The forces acting on the rod are: 1. Weight mg acting vertically downwards at the center of mass, at a distance L 2 from the lower end. 2. Normal reaction N₁ acting vertically upwards at the lower end. 3. Frictional force f acting horizontally inwards at the lower end. 4. Normal reaction N₂ from the smooth roller, acting perpendicular to the rod. The distance of the point of contact of the roller from the lower end along the rod is x = h . Taking torque about the l