Pendulum Frequency Calculator
Calculate frequency and period of pendulum oscillations using length, gravity, and initial angle
Pendulum Parameters
Distance from pivot point to center of mass
Standard Earth gravity: 9.81 m/s²
Maximum displacement from vertical position
Pendulum Frequency & Period Results
Small Angle Approximation
Valid for angles < 10°
Non-Linear Correction
Negligible correction needed
Formula used: f = (1/2π) × √(g/L)
Calculation: f = (1/2π) × √(9.81/1.000) = 0.4985 Hz
Length in meters: 1.000 m, Gravity: 9.81 m/s²
Analysis
Pendulum Formulas
Simple Pendulum
f = (1/2π) × √(g/L)
T = 2π × √(L/g)
Valid for small angles (< 10°)
Period & Frequency
T = 1/f (period in seconds)
f = 1/T (frequency in Hz)
T = time for one complete oscillation
Constants
g: gravitational acceleration
L: pendulum length
θ₀: initial angle
Common Examples
1-meter Pendulum
L = 1.0 m → f ≈ 0.5 Hz, T ≈ 2.0 s
Perfect for educational demos
Grandfather Clock
L ≈ 0.994 m → T = 2.0 s exactly
One tick per second
Foucault Pendulum
L = 67 m → T ≈ 16.4 s
Demonstrates Earth's rotation
25 cm Pendulum
L = 0.25 m → f ≈ 1.0 Hz
One oscillation per second
Pendulum Physics Tips
Mass doesn't affect frequency (in vacuum)
Longer pendulums oscillate more slowly
Higher gravity increases frequency
Large angles increase period slightly
Air resistance dampens oscillations
Understanding Pendulum Frequency and Physics
What is a Pendulum?
A pendulum is a weight (called a bob) suspended from a pivot point that can swing freely under the influence of gravity. When displaced from its equilibrium position and released, it oscillates back and forth in a predictable pattern.
Simple Harmonic Motion
For small angles (less than about 10°), a pendulum exhibits simple harmonic motion. This means the restoring force is proportional to the displacement, resulting in sinusoidal oscillations with a constant period.
Why Mass Doesn't Matter
Surprisingly, the mass of the pendulum bob doesn't affect the frequency. This is because both the gravitational force and the inertia increase proportionally with mass, canceling each other out - similar to how all objects fall at the same rate in a vacuum.
Factors Affecting Frequency
Length (L)
Primary factor. Longer pendulums have lower frequencies. Frequency is proportional to 1/√L.
Gravity (g)
Stronger gravity increases frequency. Varies slightly with altitude and latitude on Earth (9.78 to 9.83 m/s²).
Initial Angle (θ₀)
For large angles, the period increases slightly. Small angle approximation is accurate for θ₀ < 10°.
Real-World Applications
Timekeeping
Pendulum clocks use the constant period of oscillation to measure time accurately.
Historical precision: ±15 seconds/day
Seismology
Seismometers use pendulum principles to detect ground motion and earthquakes.
Can detect movements as small as nanometers
Education & Physics
Demonstrates fundamental concepts like conservation of energy and harmonic motion.
Used to measure local gravity (g)
Mathematical Foundation
Small Angle Approximation
f = (1/2π) × √(g/L)
T = 2π × √(L/g)
Valid when sin(θ) ≈ θ (θ in radians), which holds for angles less than about 10°.
Large Angle Correction
T = T₀ × [1 + (1/16)sin²(θ₀/2) + (11/3072)sin⁴(θ₀/2) + ...]
Infinite series expansion that becomes more important as the initial angle increases.