Pendulum Calculator
Calculate the period, frequency, and length of a simple pendulum. Uses the formula T = 2π√(L/g), where L is the pendulum length and g is gravitational acceleration.
The simple pendulum is one of the oldest and most instructive systems in physics. A mass on a string, swinging under gravity, has a period that depends only on the string length and gravitational acceleration — not on the mass or (for small angles) the amplitude. This remarkable isochrony was discovered by Galileo in the late 16th century, famously while watching a swinging chandelier in Pisa's cathedral. It laid the foundation for accurate timekeeping for the next 300 years.
The formula T = 2π√(L/g) is elegantly simple. Want to double the period? Quadruple the length. A 1-meter pendulum has a period of about 2 seconds — convenient for clocks where each swing equals one second tick. A grandfather clock pendulum is typically ~0.994 m for exact 2 s period. The 67-meter Foucault pendulum at the Panthéon in Paris had a period of ~16 seconds.
Pendulums also vary with gravity. On the Moon (g = 1.62 m/s²), a 1 m pendulum takes ~4.92 s per cycle — about 2.5× slower than on Earth. Geophysicists have long used pendulums to measure local g (and infer underground density variations). Pierre Bouguer's expeditions in the 1730s used pendulum timing to discover that g varies with latitude (Earth is oblate).
This calculator uses the small-angle approximation valid for swings up to ~10-15°. Larger swings introduce nonlinear corrections — period grows slightly with amplitude. For exact analysis at large amplitudes, an elliptic integral is needed.
Common applications: clock design (grandfather clocks, metronomes), accelerometers (gravity measurement), demonstrations of Earth's rotation (Foucault pendulum), seismometers, physics education, and engineering oscillator design.
Inputs
Results
Period
2.0061 s
Frequency
0.4985 Hz
Angular Freq
3.1321 rad/s
Pendulum Results
| Parameter | Value |
|---|---|
| Pendulum Length | 1 m (100.00 cm) |
| Gravity | 9.81 m/s² |
| Period (T) | 2.006067 s |
| Frequency (f) | 0.498488 Hz |
| Angular Frequency (ω) | 3.132092 rad/s |
| Oscillations/min | 29.91 |
| Formula | T = 2π√(L/g) |
Formula
How to use this calculator
- Enter pendulum length from pivot to center of mass in meters.
- Enter gravitational acceleration (default 9.81 m/s² for Earth).
- Calculator returns period, frequency, and angular frequency.
- For accurate clock design, use small amplitudes (under 10°).
- For larger amplitudes, expect period to be slightly longer than calculated.
- For exact 1 s period (seconds pendulum): L ≈ 0.248 m on Earth.
Worked examples
Grandfather clock
**Scenario:** A grandfather clock needs a 2-second period (each tick = 1 second). What length pendulum? **Calculation:** L = g × T² / (4π²) = 9.81 × 4 / 39.48 ≈ 0.994 m. **Result:** Pendulum length ~99.4 cm. Standard grandfather clocks have ~1 m pendulums. Length adjustable with a screw on the bob — clockmakers tune by raising/lowering the bob a fraction of a millimeter at a time.
Foucault pendulum
**Scenario:** Foucault's original 67-meter Pantheon pendulum. Period? **Calculation:** T = 2π × √(67/9.81) = 2π × √(6.83) = 2π × 2.61 ≈ 16.4 s. **Result:** ~16.4 seconds per cycle. Long period gives time to observe slow plane rotation (~270°/day at Paris's 49° latitude). The long, slow swing makes the rotation effect visible to the human eye over an hour or two.
Pendulum on the Moon
**Scenario:** Take a 1 m pendulum to the Moon (g = 1.62 m/s²). New period? **Calculation:** T = 2π × √(1/1.62) = 2π × √(0.617) = 2π × 0.786 ≈ 4.93 s. **Result:** ~4.93 seconds vs 2.01 s on Earth — 2.5× slower. Earth clocks brought to the Moon would run slow if not redesigned. NASA atomic clocks on Moon missions stayed accurate using cesium oscillators, not pendulums.
When to use this calculator
**Use the pendulum formula for:**
- **Clock design**: grandfather clocks, regulators, ship chronometers. - **Metronomes**: musicians' timing devices. - **Demonstrations**: physics class, science museums. - **Gravity measurement**: portable pendulum gravimeters. - **Seismic monitoring**: long-period seismometers. - **Engineering oscillators**: tuned mass dampers in skyscrapers. - **Foucault pendulums**: demonstrating Earth's rotation.
**Limitations:**
- **Small angle approximation**: errors grow for amplitudes above ~10-15°. - **Simple pendulum assumption**: requires point mass, massless string. - **Damping ignored**: real pendulums lose amplitude over time. - **Non-uniform g**: very long pendulums show g variation along length.
**Choosing a pendulum design:**
For clocks: - Length: ~1 m for 2 s period (convenient). - Bob: heavy (high momentum, less damping effect). - Suspension: knife-edge or spring (low friction). - Drive: small impulse per swing from escapement. - Compensation: bimetallic strip or invar for temperature stability.
**Foucault pendulum requirements:**
- Long: 10-70 m typical, for clear visualization. - Heavy bob: 20-200 kg, minimizes air-drag effects. - Continuous drive: electromagnetic boost to maintain amplitude. - Special suspension (gimbaled or wire): doesn't impose preferred swing plane.
**Tuned mass dampers (TMDs):**
Building-top pendulums tune to building's natural sway frequency. When wind/earthquake excites building, TMD swings out of phase, dissipating energy.
Famous examples: - Taipei 101: 660-tonne sphere, 5 m diameter, 4 stories tall. - CN Tower: 9 m TMD at top. - Citicorp Tower NYC: 400-tonne TMD.
**Gravimetric measurements:**
Pendulum period gives g: g = 4π²L / T²
Precision pendulum gravimeter: parts-per-million accuracy. Used in: - Geological surveys (oil exploration). - Geodesy (Earth's shape). - Detection of subsurface density anomalies.
Modern instruments use atom interferometry or superconducting gravimeters — even more sensitive.
**Pendulum clock history:**
- 1583: Galileo observes isochronism. - 1656: Huygens invents first pendulum clock. - 1675: Anchor escapement improves accuracy. - 1721: Graham compensation pendulum. - 1851: Foucault demonstrates Earth rotation. - 1921: Shortt clock — best mechanical timekeeper. - 1949: Atomic clocks supersede pendulums.
**Common applications:**
- **Music**: metronomes (typically 40-208 bpm). - **Sports timing**: stopwatches historically used clockwork. - **Toys**: Newton's cradle, pendulum desk ornaments. - **Demonstrations**: bowling ball pendulum showing energy conservation. - **Trapeze and swings**: same math, larger amplitudes.
**Software:**
- **MATLAB / Python**: numerical solutions including damping. - **Wolfram Mathematica**: symbolic analysis. - **Spreadsheets**: simple calculations sufficient for most uses.
**Pitfalls:**
- **Confusing length to center of mass vs string length**: matters for compound pendulums. - **Ignoring large-angle corrections**: 30° swing has 0.17% longer period. - **Forgetting damping**: real periods may have slight beat from drag. - **Using wrong g**: Earth g varies ±0.5% with latitude. - **Temperature effects**: brass pendulum lengthens 0.002%/°C; bronze 0.0018%/°C. - **Treating pendulum mass as affecting period**: it doesn't (for simple pendulum). - **Using formula for double pendulum**: complex nonlinear dynamics, often chaotic.
Common mistakes to avoid
- Forgetting the small-angle approximation breaks down above ~15°.
- Using string length instead of length to center of mass for real bobs.
- Thinking mass affects period (it doesn't for simple pendulum).
- Applying formula on rapidly rotating systems (Coriolis force matters).
- Forgetting temperature affects length (and thus period) of clock pendulums.
- Confusing period with frequency (reciprocals).
- Using simple pendulum formula for physical (rigid body) pendulums.
- Ignoring damping in real-world applications.