{
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  "tool_id": "incline-least-pull",
  "engine": "incline-least-pull",
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  "generated_at": "2026-09-23T14:21:08.541Z",
  "engine_build": "2026-09-23",
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  "failed": 0,
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      "passed": 3
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    "boundary": {
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      "passed": 4
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    "scope": [
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      "assumptions",
      "limitations",
      "sources",
      "tests"
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  "page": "https://www.calculatorx.com/calc/physics/incline-least-pull",
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  "assumptions": [],
  "limitations": [
    "The pull is β = arctan μ above the incline, so α = θ + β above the horizontal",
    "P = W (sinθ + μ cosθ) / √(1+μ²)",
    "N = W (cosθ − μ sinθ) / (1+μ²)",
    "Not a prescribed angle, tipping, a wedge, a screw, or a belt"
  ],
  "boundary_conditions": [
    "missing W, θ, or μ → MISSING_REQUIRED_INPUT",
    "W ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "θ outside [0, 90) or μ < 0 → VALUE_OUT_OF_RANGE",
    "θ + arctan μ ≥ 90° → INVALID_INPUT"
  ],
  "sources": [
    {
      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Dry friction — the least force to move a block up an incline",
      "supports": "The uphill pull is least when it is applied at the friction angle above the incline.",
      "kind": "reference"
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  ],
  "methods": [],
  "fingerprint": "sha256:bc7586251cd0d95d576ee050a65e0e52a3bdf43d7b7b5c631fc7e01523dac356",
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      "id": "golden-level",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "W": 100,
        "theta_deg": 0,
        "mu": 1
      },
      "expect": {
        "P": 70.71067811865476,
        "N": 50,
        "F": -50,
        "alpha_deg": 45,
        "beta_deg": 45,
        "model": "incline_least_pull"
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      "actual": {
        "model": "incline_least_pull",
        "W": 100,
        "theta_deg": 0,
        "mu": 1,
        "beta_deg": 45,
        "alpha_deg": 45,
        "P": 70.7106781187,
        "N": 50,
        "F": -50,
        "solver": "angled pull on an incline at the friction angle",
        "convention": "Impending motion up the plane. β = arctan μ is the angle of the pull above the incline. α = θ + β is that angle above the horizontal. P = W (sinθ + μ cosθ) / √(1+μ²). N = W (cosθ − μ sinθ) / (1+μ²). A level surface is the least-pull page. A prescribed angle is the incline-angled-pull page."
      },
      "tol": 1e-9
    },
    {
      "id": "golden-three-quarter",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "W": 100,
        "theta_deg": 0,
        "mu": 0.75
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      "expect": {
        "P": 60,
        "N": 64,
        "F": -48,
        "alpha_deg": 36.86989764584402
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      "expectWarning": null,
      "actual": {
        "model": "incline_least_pull",
        "W": 100,
        "theta_deg": 0,
        "mu": 0.75,
        "beta_deg": 36.86989764584402,
        "alpha_deg": 36.86989764584402,
        "P": 60,
        "N": 64,
        "F": -48,
        "solver": "angled pull on an incline at the friction angle",
        "convention": "Impending motion up the plane. β = arctan μ is the angle of the pull above the incline. α = θ + β is that angle above the horizontal. P = W (sinθ + μ cosθ) / √(1+μ²). N = W (cosθ − μ sinθ) / (1+μ²). A level surface is the least-pull page. A prescribed angle is the incline-angled-pull page."
      },
      "tol": 1e-9
    },
    {
      "id": "golden-smooth",
      "ok": true,
      "kind": "golden",
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        "W": 100,
        "theta_deg": 30,
        "mu": 0
      },
      "expect": {
        "P": 50,
        "N": 86.60254037844386,
        "F": 0,
        "alpha_deg": 30,
        "beta_deg": 0
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      "actual": {
        "model": "incline_least_pull",
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        "theta_deg": 30,
        "mu": 0,
        "beta_deg": 0,
        "alpha_deg": 30,
        "P": 50,
        "N": 86.6025403784,
        "F": 0,
        "solver": "angled pull on an incline at the friction angle",
        "convention": "Impending motion up the plane. β = arctan μ is the angle of the pull above the incline. α = θ + β is that angle above the horizontal. P = W (sinθ + μ cosθ) / √(1+μ²). N = W (cosθ − μ sinθ) / (1+μ²). A level surface is the least-pull page. A prescribed angle is the incline-angled-pull page."
      },
      "tol": 1e-9
    },
    {
      "id": "boundary-weight",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "W": 0,
        "theta_deg": 0,
        "mu": 1
      },
      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "Weight W must be positive."
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    {
      "id": "boundary-mu",
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      "kind": "boundary",
      "inputs": {
        "W": 100,
        "theta_deg": 0,
        "mu": -0.1
      },
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    },
    {
      "id": "boundary-steep",
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      "inputs": {
        "W": 100,
        "theta_deg": 60,
        "mu": 1
      },
      "expectError": "INVALID_INPUT",
      "code": "INVALID_INPUT",
      "error": "The friction angle measured from the horizontal is not below 90 degrees."
    },
    {
      "id": "boundary-missing",
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      "kind": "boundary",
      "inputs": {
        "W": 100,
        "theta_deg": 0
      },
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "mu is required."
    }
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  "tier": "engine_tested",
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