Simple pendulum and mass-spring oscillator calculator
Period, frequency and angular frequency of the simple pendulum and the mass-spring oscillator, with the exact amplitude correction from the elliptic integral.
The two pendulums
ten oscillations of T₀, in real time- Small oscillations
- T₀ = 2π·√(L/g) = 2π·√(1 ÷ 9.8066) = 2.0064 s
- Amplitude correction
- T = T₀ × 1.0732 = 2.1532 s
Notes (2)
Two pendulums released together at the same amplitude: one beats at the small-angle period T₀, the other at the exact period T, and the lag accumulates.
Each pendulum is drawn on a cosine of its own period; the exact pendulum’s elliptic waveform is not computed.
the small-angle rule, T₀the exact pendulum, T, and its lag
Result
continuous computation- Small-angle periodT0
- 2.0064 s
- Amplitude factorT/T0
- 1.0732
- Drift per periodT/T0 − 1
- +7.3182%
- Frequencyf
- 0.46442 Hz
- Angular frequencyω
- 2.918 rad/s
Calculation, assumptions and sources
OverviewPeriod, frequency and angular frequency of the simple pendulum (T = 2π·√(L/g), editable gravity, Earth, Moon, Mars) and the mass-spring system (T = 2π·√(m/k)).
Period, frequency and angular frequency of the simple pendulum (T = 2π·√(L/g), editable gravity, Earth, Moon, Mars) and the mass-spring system (T = 2π·√(m/k)). With the amplitude, the pendulum’s period gets the exact elliptic-integral correction: +18% at 90°.
Method & formulaspendulum: T₀ = 2π·√(L/g)
The small-angle formula assumes sin θ ≈ θ. The exact correction goes through the complete elliptic integral K, computed by the arithmetic-geometric mean using Gauss’s identity (2/π)·K(sin(θ₀/2)) = 1/agm(1, cos(θ₀/2)).
- Isochronism
- for small oscillations, the period does not depend on the amplitude (Galileo). The deviation grows with amplitude: +0.19% at 10°, +1.7% at 30°, +18% at 90°.
- Angular frequency (ω)
- the angular speed of the equivalent circular motion, ω = 2πf, in rad/s, as in x(t) = A·cos(ωt + φ).
- Spring constant (k)
- the constant of Hooke’s law F = −k·x, in newtons per metre of stretch. Two identical springs give k/2 in series and 2k in parallel.
Validity domainThe pendulum is simple: point mass, inextensible massless string, no friction.
The pendulum is simple: point mass, inextensible massless string, no friction. A massive rod or damping calls for the compound pendulum, not covered. The amplitude correction holds from 0 to 180° exclusive; at 180°, the equilibrium is unstable and the period infinite. The spring is linear (Hooke’s law) with mass negligible against m; a real spring adds roughly a third of its mass to the oscillating mass.
Period and massThe simple pendulum’s period does not depend on the mass: as in free fall, the mass that gravitates and the mass that resists cancel out.
The simple pendulum’s period does not depend on the mass: as in free fall, the mass that gravitates and the mass that resists cancel out. It depends only on L and g. The mass-spring’s period depends on the mass, not on g.
Reading the chronogram and the period chartThe chronogram marks every oscillation: a tick at n·T₀ for the small-angle rule, a tick at n·T for the exact pendulum.
The chronogram marks every oscillation: a tick at n·T₀ for the small-angle rule, a tick at n·T for the exact pendulum. The quoted lag, t/T₀ − t/T, counts in oscillations of T₀; the segment between the two rows gives it in seconds. The period chart plots the exact period T(θ₀) = T₀ / agm(1, cos(θ₀/2)) from 0 to 180° exclusive, with the small-angle period dashed and the entered amplitude as a dot.