Pendulum Period
Calculate pendulum period and frequency using T = 2π√(L/g). Enter length and gravity to get oscillation period, frequency, and angular frequency with interactive charts.
About This Calculator
The Pendulum Period Calculator computes the oscillation period, frequency, and angular frequency of a simple pendulum based on its length and the local gravitational acceleration. A simple pendulum consists of a point mass (bob) suspended from a frictionless pivot by a massless string. This tool is essential for physics students studying harmonic motion, engineers designing pendulum-based systems, and educators demonstrating oscillatory dynamics in the classroom.
The calculator uses the standard simple pendulum period formula T = 2π√(L/g), where L is the pendulum length in meters and g is the gravitational acceleration in m/s². The frequency is computed as f = 1/T, and the angular frequency as ω = 2πf = √(g/L). These formulas assume small angular displacements (less than about 15 degrees from vertical) where the motion approximates simple harmonic motion. For a 1-meter pendulum at standard Earth gravity, the period is approximately 2.006 seconds, the frequency is about 0.498 Hz, and the angular frequency is about 3.132 rad/s.
How to Use
Enter the pendulum length in meters and the gravitational acceleration in m/s² (defaults to Earth's standard gravity of 9.80665 m/s²). Click Calculate to compute the period, frequency, and angular frequency instantly. The results include an interactive bar and pie chart showing the relationship between the values, plus a detailed breakdown table for educational purposes.
Regional Notes
Worldwide: Gravitational acceleration varies slightly by geographic location. At sea level, g ranges from about 9.764 m/s² at the equator to 9.834 m/s² at the poles. The standard value of 9.80665 m/s² corresponds to approximately 45° latitude. For precision experiments, use the local g value for your location or measure it experimentally.
Educational use: Results are displayed to 4 decimal places, suitable for physics labs and classroom demonstrations. The calculator can be used to verify the relationship T² ∝ L as part of the classic pendulum experiment.
Common Pendulum Scenarios
Seconds pendulum (grandfather clock): A pendulum with a period of exactly 2 seconds requires a length of about 0.994 meters at standard Earth gravity. This is the classic seconds pendulum used in precision timekeeping since the 17th century.
Short pendulum (metronome): A metronome pendulum set to 120 BPM (0.5 second period) requires a length of about 0.062 meters (6.2 cm). Adjusting the sliding weight changes the effective length and thus the tempo.
Planetary pendulums: The same pendulum would swing at different rates on different planets. A 1-meter pendulum on the Moon (g = 1.62 m/s²) would have a period of about 4.93 seconds, while on Jupiter (g = 24.79 m/s²) the period would be about 1.26 seconds.
Frequently Asked Questions
What is the formula for the period of a simple pendulum?
The period of a simple pendulum is T = 2π√(L/g), where L is the pendulum length in meters and g is the acceleration due to gravity (9.80665 m/s² on Earth). This formula assumes small angular displacements of less than about 15 degrees.
Does the mass of the bob affect the pendulum period?
No, the mass of the bob does not affect the period of a simple pendulum. The period depends only on the pendulum length and the gravitational acceleration, as shown by the formula T = 2π√(L/g). This is because both the gravitational force and the inertial mass are proportional to the mass, which cancels out.
What is the small-angle approximation for pendulums?
The simple pendulum formula T = 2π√(L/g) is accurate within 1% for angular displacements up to 15 degrees. For larger angles, the period becomes amplitude-dependent and longer than predicted by the small-angle formula, requiring elliptic integral calculations for accurate results.
How does gravity affect the pendulum period on other planets?
Gravitational acceleration varies across celestial bodies. On Earth g = 9.80665 m/s², on the Moon g = 1.62 m/s² (period ~2.5× longer), on Mars g = 3.71 m/s² (period ~1.6× longer), and on Jupiter g = 24.79 m/s² (period ~1.6× shorter). The calculator lets you adjust gravity for any planet.
What is the relationship between pendulum period and frequency?
Period and frequency are reciprocals: f = 1/T. If a pendulum completes one full oscillation in 2 seconds, its frequency is 0.5 Hz. The angular frequency ω = 2πf = √(g/L) is also commonly used in physics to describe the oscillation rate in radians per second.
What length should a grandfather clock pendulum be?
A grandfather clock pendulum with a period of 2 seconds (one beat per second) requires a length of approximately 0.994 meters (99.4 cm) at standard Earth gravity of 9.80665 m/s². This is the classic seconds pendulum length used in precision timekeeping.
How do I measure pendulum length correctly?
Pendulum length is measured from the pivot point to the center of mass of the bob. For a simple string pendulum with a small dense bob, this is approximately the string length plus the bob's radius. Accurate length measurement is critical for precise period calculations.
Can the pendulum period be used to measure gravitational acceleration?
Yes, by rearranging the formula to g = 4π²L/T², you can calculate gravitational acceleration by measuring the pendulum length and period. This is a classic physics experiment used to determine local g values in laboratories and educational settings worldwide.