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    "P is horizontal and positive toward the uphill side",
    "Impending slip, with friction opposing the stated direction",
    "The normal is W cosθ + P sinθ",
    "Not a force parallel to the plane, tipping, a wedge, a screw, or a belt"
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      "title": "Hibbeler, Engineering Mechanics: Statics",
      "section": "Dry friction — a horizontal force on an inclined block",
      "supports": "A horizontal force resolves into P cosθ along the plane and P sinθ into the surface. Impending slip sets the friction magnitude to μN.",
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        "theta_deg": 0,
        "mu": 0.25,
        "direction": "up"
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        "P": 25,
        "N": 100,
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        "solver": "friction + math.linear_algebra solve order 2",
        "convention": "P is horizontal and positive toward the uphill side. F is positive up the plane. direction=up is impending motion up the plane, so friction acts down. The normal is W cosθ + P sinθ. This is not a force parallel to the plane."
      },
      "tol": 1e-12
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      "inputs": {
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        "theta_deg": 0,
        "mu": 0.25,
        "direction": "down"
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      "expect": {
        "P": -25,
        "N": 100,
        "F": 25,
        "direction": "down"
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        "theta_deg": 0,
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        "direction": "down",
        "P": -25,
        "N": 100,
        "F": 25,
        "solver": "friction + math.linear_algebra solve order 2",
        "convention": "P is horizontal and positive toward the uphill side. F is positive up the plane. direction=up is impending motion up the plane, so friction acts down. The normal is W cosθ + P sinθ. This is not a force parallel to the plane."
      },
      "tol": 1e-12
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        "theta_deg": 45,
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        "direction": "up"
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        "direction": "up",
        "P": 100,
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        "solver": "friction + math.linear_algebra solve order 2",
        "convention": "P is horizontal and positive toward the uphill side. F is positive up the plane. direction=up is impending motion up the plane, so friction acts down. The normal is W cosθ + P sinθ. This is not a force parallel to the plane."
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      "tol": 1e-9
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        "mu": 0.25,
        "direction": "up"
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        "mu": 0.25
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      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "direction is required."
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