Savchenko Problem 2.8.31
Physics · thermodynamics
Problem
Put a hexagon pencil on a desk with an angle of inclination > 30◦ so that it does not roll down and does not slide. What is the smallest possible angle between the pencil and the horizontal edge of the desk?Intuitive explanation
A hexagonal pencil is stable on an incline when friction and geometry work together—the key is that the pencil rests on one flat face, not rolling on an edge. As you tilt the incline, the contact face changes, and the pencil's center of mass must stay within the support polygon to prevent sliding. The critical angle occurs when the pencil is about to slip; use the geometry of the hexagon and the friction condition to find when a new face becomes the contact surface, then determine the minimum angle between pencil and desk edge.
Intuitive explanation
A hexagonal pencil is stable on an incline when friction and geometry work together—the key is that the pencil rests on one flat face, not rolling on an edge. As you tilt the incline, the contact face changes, and the pencil's center of mass must stay within the support polygon to prevent sliding. The critical angle occurs when the pencil is about to slip; use the geometry of the hexagon and the friction condition to find when a new face becomes the contact surface, then determine the minimum angle between pencil and desk edge.