Pendulum Period Calculator
Calculate the period and frequency of a simple pendulum for small and large angle oscillations
Calculate Pendulum Period
Distance from pivot point to center of mass
Standard Earth gravity: 9.81 m/s²
Pendulum Period Results
Formula used: T = 2π√(L/g)
Input values: L = 1.000 m, g = 9.81 m/s²
Note: Small angle approximation is valid for angles ≤ 15°
Period Analysis
Example Calculations
Standard Grandfather Clock
Length: 1.0 meter
Gravity: 9.81 m/s² (Earth)
Period: T = 2π√(1.0/9.81) = 2.006 seconds
Note: Perfect for "tick-tock" every second
Foucault Pendulum (Large Angle)
Length: 67 meters (Panthéon, Paris)
Initial angle: 30°
Small angle period: 16.6 seconds
Exact period: ~17.2 seconds (+3.6% correction)
Gravity Values
Pendulum Facts
Period is independent of mass and amplitude (small angles)
Longer pendulums have longer periods
Higher gravity leads to shorter periods
Large angles require exact formula for accuracy
Used in clocks, seismometers, and experiments
Understanding Pendulum Period
What is a Simple Pendulum?
A simple pendulum consists of a point mass (bob) suspended from a fixed point by a massless, inextensible string or rod. The bob oscillates back and forth under the influence of gravity, creating periodic motion.
Key Properties
- •Period depends only on length and gravity (small angles)
- •Independent of mass and initial amplitude
- •Exhibits simple harmonic motion for small angles
- •Period increases with larger initial angles
Mathematical Formulas
Small Angle Approximation (θ ≤ 15°)
T = 2π√(L/g)
Simple and accurate for small oscillations
Exact Formula (Any Angle)
T = 2π√(L/g) × K(sin(θ₀/2))
Uses elliptic integral for large angles
Note: For angles > 15°, the exact formula provides significantly more accurate results.
Historical Applications
Timekeeping
Pendulum clocks were the most accurate timekeepers for over 300 years, from their invention by Christiaan Huygens in 1656 until the 1930s.
Gravitational Studies
Pendulums were used to measure local gravitational acceleration and detect variations in Earth's gravity field across different locations.
Foucault Pendulum
Demonstrates Earth's rotation through the apparent precession of the oscillation plane, providing direct evidence of our planet's motion.
Seismology
Early seismometers used pendulum mechanisms to detect and record earthquakes and other ground vibrations.