EQUATION

Small-Angle Period Correction Series

Extends the small-angle pendulum period to larger amplitudes by adding successive correction terms: T ≈ T₀·(1 + θ₀²/16 + 11θ₀⁴/3072 + …)

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The equation

EQ.SMALL-ANGLE-PERIOD-CORRECTION

T \approx T_0\!\left(1 + \frac{\theta_0^2}{16} + \frac{11\theta_0^4}{3072}\right)

§ 02

What it solves

Extends the small-angle pendulum period to larger amplitudes by adding successive correction terms: T ≈ T₀·(1 + θ₀²/16 + 11θ₀⁴/3072 + …). Each term is positive, so the true period is always longer than the small-angle approximation.

§ 03

When to use it

When the amplitude is moderate (15°–60°) and you need more accuracy than the simple formula but do not want to evaluate the full elliptic integral. Two terms give < 0.1% error for θ₀ < 60°.

§ 04

When NOT to use it

The series diverges (actually converges slowly) for angles approaching 180°. For θ₀ > 70°, use the complete elliptic integral expression for better accuracy.

§ 05

Common mistakes

Using θ₀ in degrees instead of radians in the correction formula. Stopping at the first correction term and ignoring the second when the angle is above 30°. Applying the correction multiplicatively as a separate factor rather than multiplying the entire small-angle period by (1 + …).

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Topics that use this equation