Simple Pendulum: Period vs Length and Amplitude

Explore how pendulum length changes the period, and why amplitude matters only when the swing becomes large.

Length, L1.00 m
Amplitude, θ₀10°
Small-angle period2.006 s
Amplitude-corrected period2.010 s
Velocity magnitude, |v|0.00 m/s

Controls

1.00 m
For small angles, T ∝ √L. A longer pendulum swings more slowly.
10°
At small angles, amplitude has almost no effect. At large angles, the true period becomes longer.
Period increase from amplitude +0.19%
Current model Small-angle ≈ exact
Observe: At 10°, the small-angle approximation is extremely accurate.
Small angle: T₀ = 2π√(L/g)
Larger amplitude: T = 4√(L/g) K(sin(θ₀/2))

Pendulum motion

−10° 0° equilibrium +10° θ v

Period comparison

larger T T₀ amplitude θ₀
small-angle T₀ amplitude-corrected T selected amplitude
Predict before moving a slider

Double the length. Does the period double? Then test your prediction.