Pendulum Calculator - Oscillation Period

This physics calculator helps you calculate pendulum period, frequency, and other parameters using length and gravity. A pendulum is a mass suspended from a fixed point that swings back and forth under gravity.

Pendulum Definition

• Simple Pendulum: Point mass on massless string
• Period Formula: T = 2π√(L/g) (for small angles)
• Frequency: f = 1/T (oscillations per second)
• Angular Frequency: ω = 2πf = √(g/L)

Pendulum Formulas

• Period: T = 2π√(L/g) (simple pendulum)
• Frequency: f = 1/T = (1/2π)√(g/L)
• Angular Frequency: ω = 2πf = √(g/L)
• Maximum Velocity: v = √(2gL(1-cosθ))
• Maximum Height: h = L(1-cosθ)
• Energy Conservation: KE + PE = constant

Applications

• Timekeeping: Pendulum clocks and metronomes
• Physics Education: Simple harmonic motion
• Engineering: Vibration analysis and control
• Geophysics: Gravity measurement
• Astronomy: Foucault pendulum demonstration
• Sports: Swinging and timing

Important Notes

• Period is independent of mass (for simple pendulum)
• Period depends only on length and gravity
• Small angle approximation: sinθ ≈ θ
• Large angles require elliptic integral correction
• Energy is conserved in ideal pendulum

Pendulum Types

• Simple Pendulum: Point mass on string
• Compound Pendulum: Rigid body swinging
• Physical Pendulum: Real object with mass distribution
• Conical Pendulum: Circular motion
• Foucault Pendulum: Demonstrates Earth's rotation

Pendulum Examples

• Clock Pendulum: T = 1 s, L ≈ 0.25 m
• Metronome: Adjustable period for music
• Playground Swing: Children's entertainment
• Foucault Pendulum: Museum demonstrations
• Seismometer: Earthquake detection

Pendulum Energy

• Kinetic Energy: Maximum at bottom of swing
• Potential Energy: Maximum at highest point
• Energy Conservation: KE + PE = constant
• Damping: Energy loss due to friction
• Amplitude Decay: Gradual decrease in swing

Pendulum Corrections

• Large Angle: Period increases with amplitude
• Air Resistance: Damping reduces amplitude
• String Mass: Affects period for heavy strings
• Pivot Friction: Energy loss at suspension point
• Temperature: Length changes with temperature

Historical Significance

• Galileo Galilei: Discovered isochronism
• Christiaan Huygens: Invented pendulum clock
• Jean Foucault: Demonstrated Earth's rotation
• Timekeeping: Revolutionized time measurement
• Navigation: Improved maritime navigation