Free · Physics models
Ideal Pendulum Period with Amplitude Series
Relate a simple pendulum length, local gravity, small-angle period and frequency, with a finite-amplitude series estimate.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
Loading calculator…
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.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
A 1 m pendulum under standard gravity has small-angle period about 2.00641 s. A 30° release increases its finite-amplitude estimate.
T0 = 2π√(L/g); f0 = 1/T0. The amplitude estimate uses the captured truncated elliptic-integral series.
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
- Small-angle frequency
- 0.4984 Hz
- Small-angle period
- 2.0064 sec
- Truncated finite-amplitude period
- 2.0413 sec
Assumptions and limitations
- A point bob, massless rigid support and frictionless fixed pivot are assumed. Gravity is uniform.
- Frequency is the reciprocal of the small-angle period, not the finite-amplitude estimate. The angle is the maximum release angle from the downward vertical.
- The amplitude output is a truncated positive elliptic-integral power series, not an exact numerical integral. It loses accuracy close to a 180° release and must not be used at the unstable upright state.
- The angle is strictly between zero and 180 degrees; length and gravity must be positive.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
- Reviewed hash-pinned model correction: A point bob, massless rigid support and frictionless fixed pivot are assumed. Gravity is uniform. Frequency is the reciprocal of the small-angle period, not the finite-amplitude estimate. The angle is the maximum release angle from the downward vertical. The amplitude output is a truncated positive elliptic-integral power series, not an exact numerical integral. It loses accuracy close to a 180° release and must not be used at the unstable upright state. The angle is strictly between zero and 180 degrees; length and gravity must be positive. Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Which period determines the displayed frequency?
The frequency is the reciprocal of the small-angle period. The finite-amplitude series is shown separately.
Why avoid release angles near 180 degrees?
The exact ideal period grows without bound at the upright limit, while the finite truncated series cannot represent that divergence accurately.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
Enforce the documented finite physical branch and inverse domains.