Angle of Repose (μ = tan θ)
Worked example: μs 0.75 → 36.870° (3-4-5 slope) — press Try an example to run it live, then adjust anything.
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Grade 12Grade 12 Physics
Friction holds →
UniversityEngineering Mechanics
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Angle of Repose (μ = tan θ) explained
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.
Angle of Repose (μ = tan θ) formula
- = Coefficient of static friction
- = Angle of repose (°)
Missing one of these? Work it out first, then come back
- Coefficient of static friction — Maximum Static Friction (f = μₛN), Maximum Speed on a Flat Curve
- Angle of repose — Torque with a Lever Arm (τ = rF sin θ), Angular Velocity (ω = θ/t)