Simple Pendulum Period Calculator mechanism

Model scope

The formulas pass the physical-method check under the ideal-pendulum assumptions. Resolve the angle-unit contract and the math.js ellipticK argument convention before implementation. The exact mode should explicitly document whether K receives modulus k or parameter m=k² and should report or enforce a numerical error tolerance.

• The physical formulas are standard for an ideal simple pendulum: T_small=2π√(L/g) and T_exact=4√(L/g)K(k), with k=sin(θ0/2).
• The elliptic-integral convention is ambiguous: some libraries define K(k) using modulus k, while math.js commonly defines ellipticK(m) using parameter m in ∫0^(π/2)(1−m sin²φ)^−1/2 dφ. If math.js expects m, the implementation must pass m=sin²(θ0/2), not sin(θ0/2).
• The specification does not state that θ0 is the maximum angular displacement from vertical, nor whether input angles are radians or degrees.
• The phrase exact mode uses elliptic integral approx is ambiguous: the mathematical formula is exact for the ideal undamped pendulum, but a numerical approximation to K has a finite tolerance that is not specified.

Units and conventions

• L and g must use consistent length and time units; for L in metres, g must be in m/s² and T is in seconds.
• θ0 must be converted to radians before evaluating sin unless the interface explicitly accepts radians only.
• θ0 is assumed to be the maximum displacement angle measured from the downward vertical.
• The model assumes a point mass, massless rigid rod, no damping, no air resistance, and a fixed pivot.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "L": 1,
    "L_unit": "m",
    "g": 9.81,
    "g_unit": "m/s^2",
    "theta0": 0,
    "angle_unit": "rad"
  },
  "expected_outputs": {
    "T_small_s": 2.00606668,
    "T_exact_s": 2.00606668,
    "exact_minus_small_s": 0,
    "K_modulus": 1.57079633,
    "small_to_exact_ratio": 1
  },
  "working": "sqrt(L/g)=sqrt(1/9.81)=0.3194382825. T_small=2π(0.3194382825)=6.283185307×0.3194382825=2.00606668 s. θ0=0, so k=sin(0/2)=0 and K(0)=π/2=1.570796327. T_exact=4(0.3194382825)(π/2)=2π(0.3194382825)=2.00606668 s. Difference=2.00606668−2.00606668=0."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification exposes unsafe edge behavior unless the instrument rejects g=0 and nonphysical g values, and it has a significant angle-unit convention hazard. The small-angle branch also becomes inaccurate beyond approximately 15 degrees.

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-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json