# Pendulum Period Calculator (T = 2π√L/g)

Calculate the period and frequency of a simple pendulum using T = 2π√(L/g). Supports different gravities for Moon, Mars, and more.

## What this calculates

A simple pendulum consists of a mass (bob) suspended from a fixed point by a string or rod. For small oscillations, the period depends only on the pendulum length and gravitational acceleration: T = 2π√(L/g). Remarkably, the period is independent of the mass of the bob and (for small angles) the amplitude of the swing.

## Inputs

- **Pendulum Length (L)** (m) — min 0
- **Gravitational Acceleration (g)** (m/s²) — min 0 — Earth: 9.81, Moon: 1.62, Mars: 3.72, Jupiter: 24.79
- **Amplitude (optional)** (°) — min 0, max 90 — Formula is most accurate for small angles

## Outputs

- **Period (T)** (s) — T = 2π√(L/g): time for one complete swing
- **Frequency** (Hz) — f = 1/T: swings per second
- **Angular Frequency** (rad/s) — ω = 2π/T = √(g/L)
- **Length for 1s Period** (m) — Pendulum length needed for exactly T = 1 s

## Frequently Asked Questions

**Q: Why doesn't the mass of the pendulum affect its period?**

A: The period of a simple pendulum T = 2π√(L/g) depends only on length and gravity, not mass. This is because a heavier bob has more gravitational force pulling it, but also more inertia resisting acceleration. These two effects cancel exactly (just like in free fall), so all masses swing at the same rate for a given length.

**Q: What is the small angle approximation?**

A: The formula T = 2π√(L/g) assumes sin(θ) ≈ θ (in radians), which is accurate to within 1% for angles up to about 15°. For larger angles, the period increases. At 45°, the period is about 4% longer than the small-angle prediction. At 90°, it is about 18% longer. This calculator includes a correction term for larger amplitudes.

**Q: How long should a pendulum be for a 1-second period?**

A: On Earth (g = 9.81 m/s²), a pendulum with T = 1 second needs length L = g/(4π²) ≈ 0.2485 m (about 24.85 cm). A grandfather clock's pendulum has T = 2 seconds (1-second half-swing), requiring L ≈ 0.9941 m (about 1 meter). This is why grandfather clocks are approximately 1 meter tall.

**Q: Would a pendulum work on the Moon?**

A: Yes, but it would swing much more slowly. The Moon's gravity is about 1.62 m/s² (about 1/6 of Earth's). Since T ∝ 1/√g, the period would be √(9.81/1.62) ≈ 2.46 times longer on the Moon. A 1-second pendulum on Earth would have a period of about 2.46 seconds on the Moon.

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Source: https://vastcalc.com/calculators/physics/pendulum
Category: Physics
Last updated: 2026-04-21