Angle of Repose (μ = tan θ)

Tilt a plank until the block on it just begins to slide: at that angle the down-slope pull mg sin θ exactly equals the friction ceiling μₛ mg cos θ, the mass cancels, and μₛ = tan θ. It is the cheapest friction experiment in existence — no force gauge, no scale, just a protractor. A block that lets go at 31° reports μₛ = tan 31° ≈ 0.60.

The same angle governs bulk solids: pour sand, grain, gravel or cement and the cone stabilises at its angle of repose, roughly 34° for dry sand and 40° for crushed stone. Silo designers, mining engineers and highway embankment crews all size their structures around it, and geologists read it back out of scree slopes. The trap is dimensional intuition — since the mass cancels, a heavy block does not slide at a gentler angle than a light one of the same material, which surprises nearly everyone the first time they see it.