Large-Angle Pendulum Calculator
Enter length, gravity and initial angle to compute the exact large-angle period via the complete elliptic integral T=4√(L/g)·K(sin(θ₀/2)), the small-angle period and the % increase. L=1, θ₀=90° → T=18.6% longer than small-angle; θ₀=20° → +1.4%.
Input Data
Results
At a glance:A simple pendulum is a point mass on a massless string (or a small bob on a light rod) swinging under gravity. For tiny swings the restoring torque is τ=−mgL·sinθ, and the small-angle approximation sinθ≈θ makes the equation linear, giving T₀=2π√(L/g), a period independent of amplitude — Galileo's famous isochronism (1583) and Huygens' pendulum clock (1656). But sinθ≈θ only for small θ; with a finite initial angle θ₀ (radians) you must keep the full nonlinear term and the exact period is T=4√(L/g)·K(sin(θ₀/2)), where K is the complete elliptic integral of the first kind K(m)=∫₀^{π/2}dφ/√(1−m·sin²φ) and the modulus m=sin²(θ₀/2). √(L/g) is the small-angle time scale. Physical meaning: at larger angles the bob's path is longer and, more importantly, the restoring component mg·sinθ is smaller than mg·θ, so the motion slows and the period lengthens. The increase is non-negligible: θ₀=20° → +1.4%; θ₀=45° → +3.5%; θ₀=90° → +18.6%; as θ₀→180° the period diverges (the bob balances at the top, unstable). History: Galileo's 1583 observation that pendulum period is near-independent of amplitude; Huygens built the first pendulum clock in 1656; the elliptic-integral form was completed by Legendre in the 1820s. Applications: (1) pendulum clocks (keep θ₀ small for accuracy); (2) seismology and tiltmeters; (3) metronomes and conductors' beaters; (4) robotic and control pendulums; (5) a standard example of elliptic integrals in physics teaching.
Formula
Exact period: T = 4√(L/g)·K(sin(θ₀/2))
Small-angle: T₀ = 2π√(L/g)
Elliptic integral: K(m) = ∫₀^{π/2} dφ/√(1−m·sin²φ)
Increase: (T−T₀)/T₀ × 100%
ω₀ = √(g/L) (small-angle)
$$T = 4\sqrt{\frac{L}{g}}\,K\!\left(\sin\frac{\theta_0}{2}\right), \quad T_0 = 2\pi\sqrt{\frac{L}{g}}, \quad m = \sin^2\frac{\theta_0}{2}$$How to Use
- Enter length L (m) and gravity g (Earth 9.81).
- Enter initial angle θ₀ (°); the small-angle approx holds only for θ₀≲10°.
- The tool computes the exact period T, small-angle T₀, the increase % and ω₀.
Case Studies
Clock accuracy and large swings
A clock pendulum L=1 m, g=9.81: T₀=2π√(1/9.81)=2.007 s.
At θ₀=10° T≈2.011 s (0.2% longer); at θ₀=90° T≈2.383 s (18.6% longer).
Huygens kept θ₀ < 5° so the clock stayed within seconds/day — large swings would make it run slow.
Moon and Mars pendulums
Same L=1 m on the Moon g=1.62 → T₀=2π√(1/1.62)=4.93 s, ~2.46× the Earth period.
On Mars g=3.71 → T₀=3.26 s. A pendulum directly measures local gravity.
The large-angle correction is the same fraction regardless of g, because both T and T₀ scale as 1/√g.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.