Friction Wedge / Inclined Plane Solver mechanism

Model scope

The core equations can be accepted after the missing conventions are made explicit. The product should not claim to determine a unique force or slip direction from the current specification alone, because force-angle reference, sign mapping, coefficient selection, and unit conventions are unspecified.

• The generalized formula is standard only when the signs are correlated and the force-angle convention is defined: for impending motion up the incline, F=W(sinθ+μcosθ)/(cosφ−μsinφ); for impending motion down the incline, the corresponding signed-force relation is F=W(sinθ−μcosθ)/(cosφ+μsinφ) under a positive-upslope force convention.
• The specification does not define whether φ is measured from the incline surface, the horizontal, or the vertical; the stated formula assumes φ is measured from the incline surface.
• The specification does not define the sign convention for applied force or explicitly map each ±/∓ combination to impending upward versus downward slip.
• The angle-of-repose formula θ=atan(μ) is valid for dry Coulomb friction with no other applied force and impending slip, using the static coefficient μs; it is not generally an angle based on μk.
• The specification conflates μs and μk without stating whether the solver computes impending motion with μs or post-slip motion with μk.

Units and conventions

• W is a force, such as N or kN, not a mass; if mass is supplied, it must first be converted using W=mg.
• θ and φ must use one declared angular unit internally; the test vectors below use degrees converted to radians for trigonometric evaluation.
• θ is measured from the horizontal incline reference, and φ is measured from the incline surface.
• μ is dimensionless; use μs for impending motion and angle of repose, and μk only for motion after sliding has begun.
• The output force has the same force unit as W.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "W_N": 1000,
    "theta_deg": 30,
    "mu": 0.5,
    "phi_deg": 0,
    "branch": "impending motion up incline",
    "coefficient_type": "mu_s"
  },
  "expected_outputs": {
    "required_force_N": 933.013,
    "angle_of_repose_deg": 26.565051
  },
  "working": "θ=30°: sinθ=0.5, cosθ=0.8660254. φ=0°: cosφ=1, sinφ=0. Up-incline branch: F=W(sinθ+μcosθ)/(cosφ−μsinφ)=1000(0.5+0.5×0.8660254)/(1−0.5×0)=1000(0.9330127)=933.0127 N≈933.013 N. Angle of repose=atan(0.5)=26.565051°."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The generalized formula has a true singularity when cos(phi)-mu*sin(phi)=0, producing an undefined force that must be rejected rather than numerically evaluated. The unspecified sign convention for applied-force angle also changes the computed force from 61.4087 N to 87.1377 N for the same plausible angle input.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json