Calculator inputs
Choose a measurement mode, set gravity, and optionally apply finite-angle correction for higher-amplitude swings.
Length versus period graph
The curve shows how pendulum height changes with period under the active gravity and correction settings.
Example data table
| Case | Measured Period (s) | Frequency (Hz) | Gravity (m/s²) | Amplitude (°) | Calculated Height / Length (m) |
|---|---|---|---|---|---|
| Short pendulum | 1.00 | 1.000 | 9.80665 | 5 | 0.2484 |
| Moderate pendulum | 1.50 | 0.667 | 9.80665 | 5 | 0.5589 |
| Clock-like pendulum | 2.00 | 0.500 | 9.80665 | 5 | 0.9936 |
| Long pendulum | 3.00 | 0.333 | 9.80665 | 5 | 2.2356 |
Formula used
Small-angle period relation
T = 2π √(L / g)
Rearranged for height or effective length:
L = g × (T / 2π)2
When finite-angle correction is enabled
Tmeasured ≈ 2π √(L / g) × C
C ≈ 1 + θ²/16 + 11θ⁴/3072
L = g × (Tmeasured / (2πC))2
Here, T is the oscillation period, L is the pendulum height or effective length, g is local gravitational acceleration, and θ is the release angle in radians.
The correction factor improves estimates when the swing angle is not very small. For very large amplitudes, exact elliptic-integral methods are more accurate.
How to use this calculator
- Select the input mode: direct period, frequency, or total timed oscillations.
- Choose a gravity preset or enter a custom gravitational acceleration.
- Enter the amplitude angle and turn on correction if the swing is not tiny.
- Optionally add period uncertainty and bob mass for deeper analysis.
- Pick your preferred output unit, then press the calculate button.
- Review the result block above the form for the solved pendulum height.
- Inspect the graph, example table, and physics metrics for interpretation.
- Export the solved result as CSV or PDF for records or reports.
Frequently asked questions
1) What does the calculator mean by pendulum height?
It uses height to mean the effective pendulum length from pivot to the bob’s center of mass. That is the length required by the period formula.
2) Why does a longer pendulum have a longer period?
A longer pendulum travels through a larger arc and responds more slowly to gravity. Because period scales with the square root of length, doubling length does not double period.
3) When should I use finite-angle correction?
Use it whenever the release angle is more than roughly 10° to 15°. Larger amplitudes make the real period slightly longer than the small-angle formula predicts.
4) Can I use this for Moon or Mars gravity?
Yes. Select a preset or type a custom gravitational acceleration. Lower gravity requires a shorter pendulum for the same period, while higher gravity requires a longer one.
5) Why is period uncertainty doubled in the length estimate?
Length depends on the square of period. For small measurement errors, the relative length uncertainty is about twice the relative period uncertainty.
6) Is this suitable for a pendulum clock?
Yes. A seconds pendulum has a period near 2 seconds and an Earth length close to 0.994 meters. The calculator helps refine that value under different conditions.
7) Why does the graph change after calculation?
The graph updates to your active gravity, correction setting, and output unit. That makes the plotted curve match the exact assumptions used in your result.
8) Are height and length always identical in real experiments?
In ideal problems, yes. In real setups, effective length is measured to the bob’s center of mass, so string knots, hooks, and bob shape can shift the true value slightly.