Gravity Pendulum Calculator

Example Data Table

Formula Used

For a simple pendulum at small angles, the ideal period is: T0 = 2pi * sqrt(L / g).

Rearranging gives gravity from measured period and length: g = 4pi^2 * L / T0^2.

For moderate amplitudes, the period increases slightly. This calculator can apply a series correction: T ~= T0 * (1 + (1/16)theta0^2 + (11/3072)theta0^4 + (173/737280)theta0^6), where theta0 is in radians.

Kinematics outputs use small-angle approximations: arc amplitude A ~= L*theta0, vmax ~= omega*A, amax ~= omega^2*A. Energy uses dh = L(1 - cos theta0), so E = m*g*dh.

How to Use This Calculator

1. Pick what you want to solve for: g, T, or L.
2. Enter the other two values and choose units.
3. Enter the swing angle theta0. Keep it under 15 degrees.
4. Select a period model: large-angle series or small-angle ideal.
5. Optional: add mass to calculate energies.
6. Press Calculate. Download CSV or PDF if needed.

Gravity Pendulum Guide

A simple pendulum can estimate local gravitational acceleration using length and timing. For small angles, the period depends on the square root of length, so careful measurements can reach near 1% accuracy.

1) What the calculator solves

Choose a mode to compute gravity, period, or length. The ideal model uses T0 = 2π√(L/g). The corrected model adjusts the period for larger release angles, helping when swings exceed about 10–15°. Exported CSV and PDF tables mirror results, making lab notes consistent and shareable.

2) Typical period data

On Earth (g ≈ 9.80665 m/s²), a 1.00 m pendulum has T ≈ 2.006 s. At 0.50 m, T ≈ 1.419 s; at 2.00 m, T ≈ 2.837 s. Use these numbers to sanity‑check timing.

3) Choosing length and angle

Measure from pivot to the bob’s center of mass. A 1–2 m length produces slower motion that is easier to time. Keep amplitude under 15° to reduce nonlinearity and drag. Angle also drives the arc amplitude, vmax, and amax estimates.

4) Timing strategy and uncertainty

Time N swings and divide by N to get one period. If reaction time adds ±0.20 s uncertainty, timing 20 swings reduces period uncertainty to about ±0.01 s. This boosts computed g because g scales with 1/T².

5) Gravity varies with location

Gravity changes with latitude and altitude. Sea‑level values are roughly 9.78 to 9.83 m/s² across Earth, so a difference of 0.05 m/s² is normal. For comparison, the Moon is about 1.62 m/s² and Mars about 3.71 m/s².

6) Energy and motion outputs

With mass entered, the tool reports maximum potential energy m g L(1−cosθ0), equal to maximum kinetic energy near the bottom. For a 0.25 kg bob, L = 1 m, and 10°, the peak energy is about 0.037 J.

FAQs

Does the bob mass change the period?

No. In the simple pendulum model, period depends on length and gravity, not mass. Mass is only used here to estimate energies, which scale with m.

How many swings should I time?

Time 10–20 full swings, then divide total time by swings. More swings reduce reaction‑time error, but avoid large amplitudes that change the period.

When should I enable the angle correction?

Use the large‑angle correction when release angles are above about 10–15°. For very small angles, the ideal approximation is usually sufficient and slightly simpler.

How do I measure the pendulum length correctly?

Measure from the pivot point to the bob’s center of mass. If the bob is a sphere, measure to its center, not the string end.

Why is my gravity value different from 9.80665?

Local gravity varies with latitude, altitude, nearby geology, and measurement error. Small timing or length mistakes also affect g because it depends on 1/T².

Can I model other planets with this tool?

Yes. Enter a planet’s gravity to predict period for a chosen length, or solve for length that gives a target period. The examples in the table are Earth‑based.