Pendulum length calculator

A simple pendulum is an ideal mass on a massless string swinging under constant gravity. Rearranged for design work, the length you need for a target period is L equals g times the square of T over two pi. Students check homework clocks and metronomes; teachers and auto-graders need a deterministic oracle; lab engineers size suspension lengths before cutting string or rod stock. This calculator accepts period in seconds and gravity in metres per second squared, returns length in metres with intermediate factors for audit, and rejects non-positive period so bad timing specs never become a silent NaN. Non-positive gravity is also rejected because the model cannot produce a physical length.

How simple-pendulum length follows from the period formula

Start from the small-angle equation of motion for a point mass suspended by a fixed length under uniform gravity. Linearizing sin theta to theta yields simple harmonic motion with angular frequency omega equal to the square root of g over L. The oscillation period is therefore T equals two pi over omega, which expands to T equals two pi times the square root of L over g.

Solving for length is a single algebraic rearrangement: isolate the square root, divide both sides by two pi, square both sides, then multiply by g to obtain L equals g times the square of T over two pi, equivalently L equals g times T squared over four pi squared. The result is independent of the bob mass in the ideal model, which is why classroom labs often measure g from period and length rather than from mass.

Period must be a finite positive number of seconds; zero and negative values fail with invalid input because they are not oscillation periods. Gravity must likewise be finite and positive so length stays physical. This page intentionally uses the small-angle approximation only; large-angle elliptic integrals, physical pendulums with extended bodies, and air drag corrections are separate search intents and separate landings.

What you send and what you get back

Send two required finite numbers under the fields period and gravity. Period is the desired full-cycle time T in seconds and must be strictly greater than zero. Gravity is local gravitational acceleration g in metres per second squared and must also be strictly greater than zero; use about nine point eight one for Earth surface work, or a measured lab value when precision matters.

Classic textbook checks include a two-second period under nine point eight one metres per second squared near one metre of length, and a one-second period under the same g near one quarter metre. Keep units consistent across a multi-step lab: if you measure g in centimetres per second squared you must convert to metres per second squared before calling, or scale the length result accordingly after, so the numeric product still matches the SI interpretation your report claims.

Where pendulum length calculators show up in real work

Secondary-school and introductory university physics courses assign period-to-length drills, clock-pendulum sizing, and local-g labs that rearrange the same formula; a single trusted oracle speeds grading and demo setup. Museum and exhibit designers size long-period Foucault and demonstration pendulums from a target beat before ordering suspension hardware. Robotics and mechanism labs convert desired swing timing into arm or string length for passive dynamic walkers and low-frequency sensors as a first filter before full multibody dynamics.

Sports science and playground safety staff estimate suspension lengths for swings and trapezes from a preferred period under local gravity. If you later need period from a known length, large-angle period corrections, physical-pendulum equivalent length, or damped free decay, treat those as separate search intents and landings rather than overloading this pendulum length calculator page.