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  "assumptions": [],
  "limitations": [
    "The surface is level",
    "α = arctan μ is the angle that minimizes the pull",
    "P = μW / √(1+μ²) and N = W / (1+μ²)",
    "Not a prescribed angle, an incline, tipping, a wedge, a screw, or a belt"
  ],
  "boundary_conditions": [
    "missing W or μ → MISSING_REQUIRED_INPUT",
    "W ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "μ < 0 → VALUE_OUT_OF_RANGE"
  ],
  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Dry friction — the least force to move a block",
      "supports": "The pull P = μW / (cosα + μ sinα) is least when α = arctan μ, which gives P = μW / √(1+μ²).",
      "kind": "reference"
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  "fingerprint": "sha256:e043f55eeb07b91796a2bee9824351361301d1f4e1e7241b8cff3a0c3e1aec81",
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        "P": 70.7106781187,
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        "solver": "angled pull at the friction angle",
        "convention": "The surface is level. α = arctan μ minimizes the pull. P = μW / √(1+μ²). N = W / (1+μ²). A prescribed angle is the angled-pull page."
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        "mu": 0.75
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        "N": 64,
        "F": -48,
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        "mu": 0.75,
        "alpha_deg": 36.86989764584402,
        "P": 60,
        "N": 64,
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        "solver": "angled pull at the friction angle",
        "convention": "The surface is level. α = arctan μ minimizes the pull. P = μW / √(1+μ²). N = W / (1+μ²). A prescribed angle is the angled-pull page."
      },
      "tol": 1e-9
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        "mu": 0
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        "F": 0,
        "alpha_deg": 0
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        "F": 0,
        "solver": "angled pull at the friction angle",
        "convention": "The surface is level. α = arctan μ minimizes the pull. P = μW / √(1+μ²). N = W / (1+μ²). A prescribed angle is the angled-pull page."
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