FACT 1 — COEFFICIENT ORDERING: μ_rolling < μ_kinetic (sliding) < μ_static. Rolling friction (due to deformation) < kinetic friction (sliding surfaces) < static friction (max before motion). This ordering explains why wheels are used in transportation — rolling reduces friction. Also used in bearings. FACT 2 — SELF-ADJUSTING: Static friction adjusts itself from 0 up to f_s_max = μ_s×N. It is NOT always equal to μ_s×N; it exactly opposes the applied force until motion begins. Example: if you push gently on a heavy box that doesn't move, friction = exactly your applied force (not μ_s×N). FACT 3 — AREA INDEPENDENCE: Friction force does NOT depend on the area of contact between surfaces (for same normal force N and same surface material μ). Placing a block on its flat face (large area) vs. its thin edge (small area) → SAME friction force. This surprises many students but is a fundamental empirical law (Amontons' law). FACT 4 — NON-CONSERVATIVE: Friction converts kinetic energy to thermal energy (heat). Work done by friction depends on path length, not just the positions of start and end points. A block sliding back and forth loses energy in both directions — friction cannot restore it. Therefore friction is a non-conservative force. FACT 5 — ANGLE OF FRICTION = ANGLE OF REPOSE: Angle of friction (λ): resultant of N and limiting friction f_max makes angle λ with N. tan(λ) = f_max/N = μ_s. Angle of repose (θ): maximum angle of incline for which a block remains stationary. At angle θ: mg sinθ = μ_s mg cosθ → tanθ = μ_s. Since tan(λ) = tan(θ) = μ_s, we get λ = θ. FACT 6 — FRICTION-FORCE GRAPH: Plot friction force (y-axis) vs applied force F (x-axis). Static region: friction equals applied force — diagonal line (slope=1). At F = f_s_max: friction peaks (limiting static friction). Dynamic region: for F > f_s_max: block starts sliding; friction drops slightly to kinetic value f_k = μ_k×N and stays constant regardless of F. Key shape: straight rising line → peak → small drop → flat horizontal line. f_s_max > f_k always.

• WHY μ_rolling < μ_kinetic < μ_static: Rolling friction arises from localised deformation at the contact point (the wheel flattens slightly, then recovers). This costs much less energy than having two whole surfaces sliding against each other. Kinetic (sliding) friction is lower than static because atomic-scale cold-welding bonds are stronger when stationary (more time to form bonds) than when surfaces are sliding. Practical consequence: to skid a car, you need more force to START sliding (static) than to maintain sliding (kinetic). ABS brakes keep tyres rolling (kinetic/rolling) rather than locking (pure sliding) to maintain control.
• ANGLE OF FRICTION AND REPOSE IN PRACTICE: Both equal arctan(μ_s). If you tilt a surface and a block just starts to slide at angle θ, you have directly measured μ_s = tanθ. This is the experimental method for measuring μ_s. For a block on a vertical wall pushed by horizontal force, the angle of friction determines the minimum force needed. The angle of repose is important in geology (slope stability), engineering (storage pile angles of granular materials like sand, grain, coal).
• GRAPH TRAP — What does the flat portion represent? Once the block is sliding (kinetic region), the friction force is f_k = μ_k×N = constant regardless of how hard you push. Pushing harder does NOT increase friction on a kinetic level; it only increases acceleration. The peak (f_s_max = μ_s×N) is always higher than f_k — this is the brief moment when the block is at the threshold of motion. The dip from f_s_max to f_k represents the transition from static-dominant to kinetic-dominant friction as relative motion begins.