Pendulum Calculator

A simple pendulum — a point mass on a massless string — swings with remarkable regularity when the angle stays small. Galileo studied pendulums for timekeeping; grandfather clocks still use them today. The period depends only on string length and local gravity for small oscillations, not on bob mass or amplitude — a surprising and useful independence that makes pendulums standard in introductory mechanics labs.

Small-angle period:

T = 2π × √(L / g)

Where T is period in seconds (one full back-and-forth swing), L is pendulum length in meters, and g is gravitational acceleration. Frequency f = 1/T in hertz. Angular frequency ω = 2πf = √(g/L). The small-angle approximation holds when maximum displacement from vertical is roughly under 15°; larger swings lengthen the period slightly.

Enter length and optional g to get period and frequency. Longer pendulums swing slower — a 1 m pendulum on Earth has T ≈ 2.0 s, near one second each way. Shorter lengths raise frequency for compact sensors and seismometer designs. Physical pendulums with distributed mass use a different formula involving moment of inertia and center of mass distance.

Worked example: A lab pendulum L = 0.75 m on Earth (g = 9.81). T = 2π × √(0.75/9.81) = 2π × 0.277 ≈ 1.74 s. Frequency f = 1/1.74 ≈ 0.57 Hz. On the Moon with g = 1.62, the same length gives T = 2π × √(0.75/1.62) ≈ 4.27 s — slower swings in weaker gravity.

Connect to potential and kinetic energy for energy exchange each swing, and to the acceleration calculator for centripetal acceleration at the bob's lowest point. Pendulum timing underpins gravimetry surveys where subtle g variations reveal subsurface geology.