Simple pendulum and mass-spring oscillator
pendulum: T₀ = 2π·√(L/g)
The two coursework oscillators: the simple pendulum (T = 2π·√(L/g), editable gravity, Earth, Moon, Mars) and the mass-spring system (T = 2π·√(m/k)). The small-angle assumption is displayed, not hidden. And if you enter the amplitude, the period is corrected exactly through the elliptic integral (Gauss’s arithmetic-geometric mean): at 90° amplitude the real period is 18% longer than the textbook formula, and the tool shows it.
2.0064 s
- Calculation
- T₀ = 2π·√(L/g) = 2π·√(1 ÷ 9.8067) = 2.0064 s
T = 2.0064 s for small oscillations. The period depends neither on the mass nor on the amplitude (while it stays small), Galileo's isochronism.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulaspendulum: T₀ = 2π·√(L/g)
pendulum: T₀ = 2π·√(L/g)
mass-spring: T = 2π·√(m/k)
f = 1/T, ω = 2π/T
large amplitude: T = T₀ / agm(1, cos(θ₀/2)). Exact
Two systems, one structure: the period is 2π times the square root of an “inertia over restoring force” ratio. The pendulum ignores the mass (it sits on both sides of the equation, as in free fall); the spring ignores gravity. For amplitude, the textbook formula assumes sin θ ≈ θ; the exact correction goes through the complete elliptic integral K, evaluated here by the arithmetic-geometric mean. Gauss’s identity (2/π)·K(sin(θ₀/2)) = 1/agm(1, cos(θ₀/2)). Which converges to machine precision in a few iterations.
- Isochronism
- · for small oscillations, the period does not depend on the amplitude, Galileo’s observation that founded clockmaking. It stops being true at large amplitudes: +0.05% at 10°, +1.7% at 30°, +18% at 90°.
- Angular frequency (ω)
- · the angular speed of the equivalent circular motion, in rad/s: ω = 2πf. It is what appears in x(t) = A·cos(ωt + φ).
- Spring constant (k)
- · the proportionality constant of Hooke’s law F = −k·x, how hard the spring pulls per metre of stretch. Two identical springs in series: k/2; in parallel: 2k.
Validity domainThe pendulum is modelled simple: point mass, inextensible massless string.
The pendulum is modelled simple: point mass, inextensible massless string. No friction, a real pendulum (massive rod, damping) requires the compound pendulum, outside this tool. The amplitude correction is exact for this model, from 0 to 180° exclusive (at 180°, unstable equilibrium, infinite period). The spring is assumed linear (Hooke’s law) with mass negligible against m; a real spring adds roughly a third of its own mass to the oscillation.
Common pitfall: making the pendulum heavier changes nothingThe simple pendulum’s period does not depend on the mass, like free fall.
The simple pendulum’s period does not depend on the mass, like free fall. And for the same reason: mass both gravitates and resists, and cancels out. It depends only on L and g, which is how g used to be measured. The mass-spring does exactly the opposite: its period depends on the mass but not on g, hanging or horizontal, on Earth or on the Moon, it beats the same. Mixing up the two dependencies is the archetypal exam mistake.