Free · Physics models
Pendulum Period with Finite-Amplitude Approximation
Compare a simple pendulum’s small-angle frequency with a finite-amplitude series approximation through sin⁶(θ/2).
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
period = 2.0064093 sec; frequency = 0.49840280 Hz; freq2 = 0.48987543 Hz.
T₀ = 2π√(L/g); T ≈ T₀ [1+k²/4+9k⁴/64+25k⁶/256], k = sin(θ/2).
Worked example
Enter these known values and leave the other values blank.
- Acceleration of gravity (g)
- 1 g
- Pendulum length (L)
- 1 m
- Initial angle (θ₀)
- 30 deg
- Pendulum period (T)
- 2.0064 sec
- Pendulum frequency (f) — small angles
- 0.4984 Hz
- Pendulum frequency (f) — nonlinear
- 0.4899 Hz
Assumptions and limitations
- Compare a simple pendulum’s small-angle frequency with a finite-amplitude series approximation through sin⁶(θ/2).
- Use a point mass, massless rigid support and uniform gravity. Angle is the maximum displacement from downward vertical, not the instantaneous angle.
- The retained angle range is greater than zero and less than 180°. The truncated series is useful at moderate amplitudes; use it cautiously above 90°. It underestimates the full elliptic-integral period near 180°, where the true period diverges.
- Inputs and inverse outputs share the displayed unit selectors. Clear a known field to solve for it, and lock quantities that must remain fixed. Singular inverses may have no finite result.
Common questions
Why does frequency decrease with amplitude?
At larger swings the exact period increases; the positive correction terms approximate that effect.
Does mass change the period?
Not for the ideal simple pendulum represented here.
References
Compare a simple pendulum’s small-angle frequency with a finite-amplitude series approximation through sin⁶(θ/2). Use a point mass, massless rigid support and uniform gravity. Angle is the maximum displacement from downward vertical, not the instantaneous angle. The truncated series is useful at moderate amplitudes. It underestimates the full elliptic-integral period near 180°, where the true period diverges. Restrict amplitude to less than 90° for this estimate.