{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"pendulum"},"result":{"entity_type":"calculator","id":"pendulum","calculator_id":"pendulum","canonical_url":"https://tttkmbb.com/physics/pendulum","name":"Pendulum Period Calculator","title":"Simple Pendulum Calculator – Period T = 2π√(L/g), Frequency and Large-Amplitude Correction","category":"physics","category_name":"Physics","tool_name":"calculate_pendulum_period","featured_mcp_tool":false,"description":"Computes the period, frequency and angular frequency of a simple pendulum from its length with the small-angle formula T = 2π√(L/g); when an amplitude is given it also returns the exact period from the arithmetic–geometric mean.","use_when":"You need how long a pendulum of known length takes to swing on Earth or another body, or how much a large swing angle lengthens the period.","do_not_use_when":"The bob's size or the rod's mass is not negligible (physical pendulum, T = 2π√(I / (m·g·d))), or the oscillator is a mass on a spring (use hookes-law).","inputs":[{"name":"length_m","label":"Pendulum length","type":"number","unit":"m","required":true,"exclusive_min":0,"description":"Distance from the pivot to the centre of mass of the bob, in metres.","example":1},{"name":"gravity_m_s2","label":"Gravitational acceleration","type":"number","unit":"m/s²","required":false,"default":9.80665,"max":10000,"exclusive_min":0,"description":"Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8).","example":9.80665},{"name":"amplitude_degrees","label":"Amplitude","type":"number","unit":"°","required":false,"min":0,"max":170,"description":"Optional maximum swing angle from vertical; enables the exact (large-amplitude) period.","example":30}],"outputs":[{"name":"period_s","label":"Period (small-angle)","type":"number","unit":"s","decimals":4,"description":"T = 2π√(L/g), one full back-and-forth swing."},{"name":"frequency_hz","label":"Frequency","type":"number","unit":"Hz","decimals":4,"description":"f = 1 / T."},{"name":"angular_frequency_rad_s","label":"Angular frequency","type":"number","unit":"rad/s","decimals":4,"description":"ω = √(g / L)."},{"name":"period_at_amplitude_s","label":"Period at given amplitude","type":"number","unit":"s","decimals":4,"description":"Exact period T / agm(1, cos(θ0/2)) (only when amplitude_degrees is given)."},{"name":"amplitude_correction_percent","label":"Correction","type":"number","unit":"%","decimals":4,"description":"How much longer the exact period is than the small-angle value (only when amplitude_degrees is given)."}],"input_schema":{"type":"object","properties":{"length_m":{"description":"Distance from the pivot to the centre of mass of the bob, in metres. Unit: m.","type":"number","exclusiveMinimum":0,"examples":[1],"x-unit":"m"},"gravity_m_s2":{"description":"Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8). Unit: m/s².","type":"number","maximum":10000,"exclusiveMinimum":0,"default":9.80665,"examples":[9.80665],"x-unit":"m/s²"},"amplitude_degrees":{"description":"Optional maximum swing angle from vertical; enables the exact (large-amplitude) period. Unit: °.","type":"number","minimum":0,"maximum":170,"examples":[30],"x-unit":"°"}},"additionalProperties":false,"required":["length_m"]},"output_schema":{"type":"object","properties":{"period_s":{"description":"T = 2π√(L/g), one full back-and-forth swing. Unit: s.","type":"number","x-unit":"s"},"frequency_hz":{"description":"f = 1 / T. Unit: Hz.","type":"number","x-unit":"Hz"},"angular_frequency_rad_s":{"description":"ω = √(g / L). Unit: rad/s.","type":"number","x-unit":"rad/s"},"period_at_amplitude_s":{"description":"Exact period T / agm(1, cos(θ0/2)) (only when amplitude_degrees is given). Unit: s.","type":"number","x-unit":"s"},"amplitude_correction_percent":{"description":"How much longer the exact period is than the small-angle value (only when amplitude_degrees is given). Unit: %.","type":"number","x-unit":"%"}}},"formula":"T = 2π × √(length_m / gravity_m_s2); f = 1 / T; ω = √(gravity_m_s2 / length_m); exact period for amplitude θ0: T_exact = T / agm(1, cos(θ0 / 2)) (arithmetic–geometric mean)","method":"The small-angle formula is within 0.1 % up to about 7° amplitude, 1.7 % low at 30° and 18 % low at 90°; the AGM result is exact for a point mass on a massless rod without friction.","sources":[{"name":"HyperPhysics – Simple pendulum","url":"http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html","type":"reference","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Pendulum (mechanics)","url":"https://en.wikipedia.org/wiki/Pendulum_(mechanics)","type":"reference","retrieved_at":"2026-09-23"},{"name":"NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s²","url":"https://physics.nist.gov/cgi-bin/cuu/Value?gn","type":"standard","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"1 m pendulum, 30° amplitude","inputs":{"length_m":1,"amplitude_degrees":30},"expected":{"period_s":2.0064,"frequency_hz":0.4984,"angular_frequency_rad_s":3.1316,"period_at_amplitude_s":2.0413,"amplitude_correction_percent":1.7409},"url":"https://tttkmbb.com/api/v1/calculate/pendulum?length_m=1&amplitude_degrees=30"},{"name":"0.25 m pendulum","inputs":{"length_m":0.25},"expected":{"period_s":1.0032,"frequency_hz":0.9968},"url":"https://tttkmbb.com/api/v1/calculate/pendulum?length_m=0.25"}],"faq":[{"q":"Does the mass of the bob change the period?","a":"No. For a simple pendulum the period depends only on length and gravity; mass cancels out."},{"q":"What length gives a period of exactly 2 s?","a":"About 0.994 m at g = 9.80665 m/s² (L = g·T² / 4π²), the classic 'seconds pendulum' with a one-second half swing."}],"tags":["pendulum","pendulum period","simple harmonic motion","frequency","seconds pendulum"],"related":[{"calculator_id":"hookes-law","reason":"Spring–mass oscillator period."},{"calculator_id":"free-fall","reason":"Another way to measure g."},{"calculator_id":"wavelength-frequency","reason":"Frequency and period of waves."}],"links":{"html":"https://tttkmbb.com/physics/pendulum","markdown":"https://tttkmbb.com/physics/pendulum.md","json":"https://tttkmbb.com/physics/pendulum.json","api":"https://tttkmbb.com/api/v1/calculate/pendulum","schema":"https://tttkmbb.com/api/v1/calculators/pendulum","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:23:00Z"}