Pendulum Calculator API

Gravity-driven pendulum maths as an API, computed locally and deterministically. The simple endpoint computes the period of a simple pendulum, T = 2π·√(L/g), together with its frequency and angular frequency, and solves for the length needed to give a target period — with an optional large-amplitude correction (the first two terms of the amplitude series) for swings where the small-angle approximation no longer holds. The physical endpoint handles a compound (physical) pendulum — any rigid body swinging about a pivot — from its moment of inertia about the pivot, its mass and the distance from the pivot to its centre of mass, T = 2π·√(I/(m·g·d)), and reports the equivalent simple-pendulum length I/(m·d). The conical endpoint solves a conical pendulum, a bob sweeping a horizontal circle, T = 2π·√(L·cosθ/g), giving the radius of the circle, the speed of the bob, the angular velocity and — with a mass — the string tension m·g/cosθ and the centripetal force. Everything is an idealised system under constant gravity with no air resistance or string mass, computed locally and deterministically, so it is instant and private. Ideal for physics-education and engineering tools, clock and metronome design, swing and amusement-ride dynamics, and STEM teaching. Pure local computation — no key, no third-party service, instant. Live, nothing stored. 3 endpoints. This is gravity-pendulum dynamics; for spring-mass-damper vibration use a vibration API, for rotational kinetic energy use a flywheel API.