# Pendulum Period Calculator (T = 2π√(L/g)) Compute period, length, or gravity of a simple pendulum, plus amplitude-based peak speed and energy > Canonical page: https://elysiatools.com/en/tools/pendulum-calculator - **Category:** Math & Numbers - **Keywords:** pendulum, simple pendulum, period, oscillation, gravity, length, frequency, angular frequency, amplitude, physics, mechanics, harmonic motion ## Overview A simple pendulum is a mass on a string of length L swinging under gravity g. Small-angle equations (given any two of T, L, g — the third follows): - Period: T = 2π · √(L / g) - Length: L = g · (T / 2π)² - Gravity: g = 4π² · L / T² - Frequency: f = 1 / T With amplitude θ₀: h = L(1 − cos θ₀), v_max = √(2gh), a_max = g·sin θ₀. Small-angle approximation: accurate within 0.2% at θ₀ ≤ 15°. g defaults to 9.80665 m/s². ## Inputs - **Solve For** (select) - **Period (T, s)** (number): leave blank when solving for period - **Length (L, m)** (number): leave blank when solving for length - **Gravity (g, m/s²)** (number): 9.80665 (Earth). Moon ≈ 1.62, Mars ≈ 3.71 - **Amplitude (θ₀, °)** (number): optional, e.g. 10 (small-angle approx.) - **Bob Mass (m, kg)** (number): optional, for kinetic energy ## When to use - When designing or calibrating mechanical pendulum clocks to determine the exact rod length required for a specific swing interval. - When calculating local gravitational acceleration in physics experiments by measuring a pendulum's length and oscillation period. - When analyzing the maximum velocity, peak acceleration, and energy of a swinging mass at a specific angular amplitude. ## How it works - Select the target variable you want to calculate: Period (T), Length (L), or Gravity (g). - Enter the known parameters, such as the pendulum length in meters, period in seconds, or local gravitational acceleration. - Optionally input the angular amplitude in degrees and the bob mass in kilograms to compute peak speed, height, and kinetic energy. - Click calculate to instantly generate the solved variable, frequency, and amplitude-based motion dynamics. ## Use cases - Physics students verifying experimental gravity measurements using a string, a weight, and a stopwatch. - Horologists calculating the exact pendulum length needed for a clock to tick at precise one-second intervals. - Engineers analyzing the maximum tension and velocity of suspended swinging loads in mechanical systems. ## Frequently asked questions ### What is the small-angle approximation limit for this calculator? The small-angle approximation is highly accurate, keeping errors under 0.2% for swing amplitudes of 15 degrees or less. ### How does the calculator find the maximum speed of the pendulum bob? It uses conservation of energy, calculating the height change from the input amplitude and length to find the peak velocity at the lowest point. ### Can I use this calculator to find gravity on other planets? Yes. Select 'Gravity (g)' as the target variable and input the pendulum's length and measured period to find the local gravitational acceleration. ### Does the mass of the bob affect the pendulum's period? No. The period of a simple pendulum depends solely on its length and gravity. Mass is only used to calculate kinetic energy. ### What units of measurement should I use? Input length in meters (m), period in seconds (s), gravity in meters per second squared (m/s²), amplitude in degrees (°), and mass in kilograms (kg). ## Related tools - [Projectile Motion Calculator (Range, Height, Time)](https://elysiatools.com/en/tools/projectile-motion-calculator): Compute range, max height, flight time, and the full trajectory of a projectile given speed, angle, and optional initial height - [Debt Payoff Calculator](https://elysiatools.com/en/tools/debt-payoff-calculator): General-purpose debt payoff planner comparing avalanche, snowball, and minimum-only strategies. Enter debts as 'name, balance, APR%, minPayment $' per line plus an extra monthly payment. 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