Simple Pendulum
A simple pendulum consists of a point mass suspended from a fixed point by a massless string. For small angles, the motion is simple harmonic with period T = 2π√(L/g), independent of mass and amplitude.
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Period is independent of mass—heavier and lighter bobs swing at the same rate. Doubling length increases period by √2 ≈ 1.41×. A seconds pendulum (T=2s) has length ≈ 0.994 m on Earth. Foucault pendulums demonstrate Earth's rotation; the swing plane appears to rotate.
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Why: Pendulum physics underpins timekeeping (clocks, metronomes), gravimetry, and demonstrations of Earth's rotation (Foucault pendulum). Understanding period-length relationships is essential for mechanical oscillators.
How: The period depends only on length and gravity: T = 2π√(L/g). Mass and amplitude (for small angles) do not affect the period. Energy alternates between kinetic and potential.
Run the calculator when you are ready.
🕰️ Grandfather Clock
1m pendulum for 2-second period
🎢 Playground Swing
3m chain swing
🎵 Metronome
Period = 0.5s (120 BPM)
🌍 Foucault Pendulum
67m museum pendulum
🌙 Pendulum on Moon
1m pendulum, g = 1.62 m/s²
🔬 Physics Lab
50cm lab pendulum
🏗️ Wrecking Ball
20m cable, 500kg ball
💎 Earring Swing
3cm dangling earring
🎡 Pendulum Ride
15m amusement ride
⏱️ Seconds Pendulum
T = 2s exactly (L ≈ 0.994m)
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For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Galileo discovered pendulum isochronism in 1583 by observing a chandelier in Pisa Cathedral.
— History of Physics
Christiaan Huygens built the first pendulum clock in 1656, revolutionizing timekeeping.
— Horology
Foucault's 1851 pendulum in Paris demonstrated Earth's rotation without external reference.
— Classical Mechanics
For angles <15°, the small-angle approximation sin(θ)≈θ holds with <1% error.
— Harmonic Motion
What is a Simple Pendulum?
A simple pendulum consists of a point mass suspended from a fixed point by a massless, inextensible string. When displaced from equilibrium and released, it oscillates in a vertical plane under gravity. For small angles, the motion is simple harmonic with a period that depends only on length and gravity—not on mass or amplitude.
Period Independence
The period depends only on length (L) and gravity (g), not on mass or amplitude (for small angles).
T = 2π√(L/g)
Small Angle Approximation
For angles less than ~15°, sin(θ) ≈ θ, making the motion truly simple harmonic.
At 15°: sin(15°) = 0.259, θ = 0.262 rad
Error: ~1%
Energy Conservation
Total mechanical energy is conserved (ignoring friction). PE and KE continuously interchange.
E = PE + KE = constant
How to Calculate Pendulum Properties
🧮 Period & Frequency
Period
Frequency
Angular Frequency
📊 Position & Velocity
Angular Position
Angular Velocity
Max Linear Velocity
Applications of Simple Pendulums
🕰️ Timekeeping
Grandfather clocks, metronomes, and early scientific instruments used pendulum regularity for precise timing.
🌍 Measuring Gravity
Gravimeters use pendulum period to detect gravity variations for geological surveys and mineral exploration.
🔬 Physics Education
Classic experiment for demonstrating harmonic motion, energy conservation, and period-length relationships.
🌐 Earth Rotation
Foucault pendulums demonstrate Earth's rotation as the swing plane appears to rotate over hours.
🏗️ Construction
Plumb bobs and construction pendulums establish vertical reference lines for accurate building.
🎢 Amusement Rides
Pendulum rides use the physics of oscillation for thrilling but predictable motion.
Complete Formula Reference
Period
Length from Period
Max Velocity
Max Height
String Tension (max)
Total Energy
Large Angle Corrections
For larger angles, the simple formula T = 2π√(L/g) underestimates the period. A more accurate formula includes correction terms:
| Initial Angle | Period Error (simple formula) |
|---|---|
| 5° | 0.05% |
| 15° | 0.5% |
| 30° | 1.7% |
| 45° | 4.0% |
| 60° | 7.3% |
| 90° | 18% |
Frequently Asked Questions
Why doesn't mass affect the period?
The gravitational force (mg) and the inertial resistance (ma) both depend on mass. They cancel out in the equation of motion, leaving period independent of mass—just like free fall!
How does a grandfather clock keep accurate time?
The pendulum is adjusted to exactly 1 meter (for T = 2s). An escapement mechanism gives the pendulum small impulses to overcome friction, maintaining constant amplitude without affecting period.
Would a pendulum work on the Moon?
Yes, but with a longer period! Moon's gravity is ~1/6 of Earth's, so T_moon = T_earth × √6 ≈ 2.45 times longer. A seconds pendulum (T=2s on Earth) would have T≈4.9s on the Moon.
Why does a Foucault pendulum appear to rotate?
It doesn't actually rotate—the Earth rotates beneath it! The swing plane stays fixed relative to the stars while Earth turns. At the poles, it completes one apparent rotation in 24 hours.
Reference: Pendulum Lengths for Common Periods
| Period (s) | Length (m) | Length (cm) | Application |
|---|---|---|---|
| 0.5 | 0.062 | 6.2 | Fast metronome (120 BPM) |
| 1.0 | 0.248 | 24.8 | Quick pendulum |
| 2.0 | 0.994 | 99.4 | Seconds pendulum (clock) |
| 3.0 | 2.236 | 223.6 | Large clock |
| 5.0 | 6.21 | 621 | Slow swing |
| 10.0 | 24.85 | 2485 | Large Foucault pendulum |
Tips and Common Mistakes
✅ Best Practices
- • Keep initial angle small (<15°) for accuracy
- • Measure length to center of mass of bob
- • Time multiple swings and divide
- • Ensure pivot has minimal friction
❌ Common Mistakes
- • Using large angles without correction
- • Measuring length to bottom of bob
- • Ignoring string mass for accuracy
- • Forgetting period is independent of mass
Practice Problems
Problem 1: Clock Pendulum
What length pendulum gives exactly T = 1 second?
L = gT²/(4π²) = 9.81 × 1²/(4π²) = 0.248 m = 24.8 cm
Problem 2: Maximum Speed
A 1m pendulum starts at 10°. What's its maximum speed?
v_max = √(2gL(1-cosθ₀)) = √(2 × 9.81 × 1 × (1-cos10°))
v_max = √(19.62 × 0.0152) = 0.546 m/s
Problem 3: Planetary Pendulum
A pendulum has T = 2s on Earth. What's its period on Mars (g = 3.72 m/s²)?
T ∝ 1/√g, so T_Mars = T_Earth × √(g_Earth/g_Mars)
T_Mars = 2 × √(9.81/3.72) = 2 × 1.624 = 3.25 seconds
Historical Significance
🔬 Galileo Galilei (1583)
At age 19, Galileo noticed that a swinging chandelier in the Pisa cathedral took the same time per swing regardless of amplitude. This observation led to the discovery of pendulum isochronism.
⏰ Christiaan Huygens (1656)
Built the first practical pendulum clock, which revolutionized timekeeping. His work on pendulum theory established the mathematics of oscillatory motion.
🌍 Léon Foucault (1851)
Demonstrated Earth's rotation using a 67m pendulum in the Panthéon, Paris. The swing plane appeared to rotate as Earth turned beneath it.
📐 Henry Kater (1817)
Invented the reversible pendulum for precise measurement of gravitational acceleration, which remained the standard method for over a century.
Pendulum Periods on Different Celestial Bodies
| Body | g (m/s²) | Period (1m pendulum) | Relative to Earth |
|---|---|---|---|
| Sun (surface) | 274 | 0.38 s | 0.19× |
| Mercury | 3.70 | 3.27 s | 1.63× |
| Venus | 8.87 | 2.11 s | 1.05× |
| Earth | 9.81 | 2.01 s | 1.00× |
| Moon | 1.62 | 4.93 s | 2.46× |
| Mars | 3.72 | 3.26 s | 1.62× |
| Jupiter | 24.79 | 1.26 s | 0.63× |
Derivation of Pendulum Motion
Step 1: Restoring Force
The component of gravity tangent to the arc provides the restoring force:
Step 2: Newton's Second Law
For arc length s = Lθ and acceleration d²s/dt² = L d²θ/dt²:
Step 3: Small Angle Approximation
For small θ, sin(θ) ≈ θ, giving simple harmonic motion:
Key Relationships Summary
Double length
1.41× period
T ∝ √L
Quadruple length
2× period
T ∝ √L
Double gravity
0.71× period
T ∝ 1/√g
Double mass
Same period!
T independent of m
Real Pendulums: Damping Effects
Real pendulums lose energy to air resistance and friction at the pivot. The amplitude decreases exponentially over time, though the period remains nearly constant for light damping.
Damped Motion
Amplitude decreases over time:
ω' = √(ω₀² - γ²)
Quality Factor
Measures energy loss per cycle:
Higher Q = less damping
Frequently Asked Questions
Why doesn't mass affect the period of a simple pendulum?
The gravitational force (mg) and the inertial resistance (ma) both depend on mass. They cancel out in the equation of motion, leaving period independent of mass—just like free fall! This is a fundamental property of simple harmonic motion under gravity.
How does a grandfather clock keep accurate time?
The pendulum is adjusted to exactly 1 meter (for T = 2s). An escapement mechanism gives the pendulum small impulses to overcome friction, maintaining constant amplitude without affecting period. The period depends only on length and gravity, making it very stable.
Would a pendulum work on the Moon?
Yes, but with a longer period! Moon's gravity is ~1/6 of Earth's, so T_moon = T_earth × √6 ≈ 2.45 times longer. A seconds pendulum (T=2s on Earth) would have T≈4.9s on the Moon. The pendulum would swing much slower but still oscillate.
Why does a Foucault pendulum appear to rotate?
It doesn't actually rotate—the Earth rotates beneath it! The swing plane stays fixed relative to the stars while Earth turns. At the poles, it completes one apparent rotation in 24 hours. This demonstrates Earth's rotation without external reference.
What is the small angle approximation and when does it fail?
For angles less than ~15°, sin(θ) ≈ θ, making the motion truly simple harmonic. At 15°, the error is only ~1%. For larger angles, the period increases. At 30°, error is ~1.7%; at 60°, it's ~7.3%. The simple formula T = 2π√(L/g) underestimates period for large angles.
How do you measure gravitational acceleration using a pendulum?
Measure the period T and length L precisely, then use g = 4π²L/T². This is the principle behind gravimeters used in geological surveys. The Kater reversible pendulum was the standard method for measuring g for over a century, achieving accuracies better than 0.01%.
What causes damping in real pendulums?
Real pendulums lose energy to air resistance (proportional to velocity squared) and friction at the pivot (proportional to velocity). The amplitude decreases exponentially: A(t) = A₀e^(-γt). The quality factor Q = ω₀/(2γ) measures how many oscillations before significant decay.
📚 Official Data Sources
NIST Physical Measurement Laboratory
US National Institute of Standards - Physical constants and measurements
Last Updated: 2026-02-01
MIT OpenCourseWare Physics
MIT physics courses including harmonic motion and pendulums
Last Updated: 2025-12-15
Physics Hypertextbook
Comprehensive physics reference including pendulum mechanics
Last Updated: 2025-11-20
HyperPhysics (Georgia State)
Georgia State University physics reference - pendulum oscillations
Last Updated: 2025-10-10
⚠️ Disclaimer: This calculator provides theoretical estimates based on the simple pendulum formula T = 2π√(L/g), which assumes small angles, massless string, point mass bob, and no damping. Actual pendulum behavior may vary due to air resistance, string mass, pivot friction, large-angle effects, and environmental factors. For angles greater than 15°, corrections are needed. Always verify critical measurements with professional instruments. This calculator is for educational and planning purposes only.
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