Pendulum Energy
A simple pendulum oscillates with constant total mechanical energy. KE + PE = constant; energy transfers between kinetic (bottom) and potential (extremes).
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Total E = mgL(1−cos θ_max) constant v_max = √(2gL(1−cos θ_max)) at bottom Small angle: T = 2π√(L/g) independent of amplitude Energy oscillates between KE and PE
Ready to run the numbers?
Why: Pendulum energy conservation explains why amplitude determines period and why velocity is maximum at bottom.
How: PE = mgh with h = L(1−cos θ). KE = ½mv². Total E constant. At bottom: all KE; at extremes: all PE.
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Input Parameters
Visualizations
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Period T = 2π√(L/g) for small angles
— Harmonic Motion
Maximum velocity at bottom of swing
— Energy Conservation
PE = mgh where h = L(1−cos θ)
— Geometry
Total mechanical energy conserved (no friction)
— Classical Mechanics
What is Pendulum Energy?
A swinging pendulum continuously exchanges kinetic energy (energy of motion) and potential energy (energy of position). At the extremes of swing, all energy is potential; at the lowest point, all energy is kinetic. This energy transformation demonstrates the fundamental principle of conservation of mechanical energy in an ideal system.
Kinetic Energy (KE)
- • Maximum at bottom (v = v_max)
- • Zero at extremes (v = 0)
- • Depends on velocity squared
Potential Energy (PE)
- • Maximum at extremes (h = h_max)
- • Zero at bottom (h = 0)
- • Depends on height above lowest point
Essential Formulas
Height from Angle
Max Velocity
Energy Conservation
Velocity at Any Angle
Period
Small angle approximation
Total Energy
Energy at Key Points
| Position | Height | Velocity | KE | PE |
|---|---|---|---|---|
| Extreme (max θ) | h_max | 0 | 0 | E_total |
| Midway | h_max/2 | v_max/√2 | E/2 | E/2 |
| Bottom (θ = 0) | 0 | v_max | E_total | 0 |
Practice Problems
Problem 1
A 2 kg pendulum bob swings on a 1.5 m string, released from 45°. Find the speed at the bottom.
h = L(1 - cos 45°) = 1.5(1 - 0.707) = 0.44 m
v = √(2gh) = √(2 × 9.81 × 0.44) = 2.94 m/s
Problem 2
A pendulum passes through equilibrium at 3 m/s. If L = 1 m, find the maximum angle.
h = v²/(2g) = 9/(2 × 9.81) = 0.459 m
cos θ = 1 - h/L = 1 - 0.459 = 0.541
θ = cos⁻¹(0.541) = 57.2°
Key Takeaways
- •Pendulum energy continuously transforms between kinetic (KE = ½mv²) and potential (PE = mgh) forms, with total mechanical energy conserved in ideal systems.
- •Maximum kinetic energy occurs at the lowest point (bottom) where velocity is maximum, while maximum potential energy occurs at the extremes where height is maximum.
- •Height above lowest point relates to angle: h = L(1 - cos θ), where L is pendulum length and θ is deflection angle from vertical.
- •Maximum velocity is independent of mass: v_max = √(2gh_max), but total energy depends on mass: E_total = mgh_max.
- •Energy conservation principle: E_total = KE + PE = constant (in ideal frictionless system), demonstrating fundamental physics conservation laws.
- •Period depends only on length and gravity: T = 2π√(L/g) for small angles, independent of mass and amplitude.
Did You Know?
🕐 The Foucault pendulum at the Pantheon in Paris demonstrates Earth's rotation. Its 67-meter length and 28 kg mass create a 16.5-second period, with the plane of oscillation rotating 11.3° per hour due to the Coriolis effect.
Source: HyperPhysics - Pendulum
⚖️ Galileo discovered that pendulum period is independent of amplitude (for small angles) and mass, leading to the isochronism principle. This discovery was crucial for developing accurate timekeeping devices.
Source: MIT Classical Mechanics
🎢 Amusement park swings can reach speeds over 30 m/s (67 mph) at the bottom of their arc. A 70 kg person on a 20 m swing released from 60° has kinetic energy of over 20,000 J at the bottom!
Source: Physics Classroom
🔔 Church bell clappers are pendulums that transfer kinetic energy to sound. A 500 kg bell with a 0.8 m clapper can generate over 1000 J of kinetic energy, creating sound waves that travel kilometers.
Source: Khan Academy
Expert Tips
- 💡For small angles (< 15°), the period formula T = 2π√(L/g) is accurate within 1%. For larger angles, use the complete elliptic integral solution for precise calculations.
- 💡Always set potential energy reference at the lowest point (h = 0) for consistency. This makes calculations simpler and ensures PE = 0 at maximum velocity.
- 💡For real pendulums, account for energy losses due to air resistance and pivot friction. Energy decreases exponentially: E(t) = E₀e^(-γt), where γ is the damping coefficient.
- 💡Maximum velocity occurs at the bottom regardless of release angle. Use v_max = √(2gh_max) = √(2gL(1-cos θ_max)) for quick calculations.
- 💡For compound pendulums (physical pendulums), use the center of mass distance and moment of inertia. The period becomes T = 2π√(I/(mgd)), where I is moment of inertia and d is distance to center of mass.
- 💡Energy conservation provides a powerful check: at any point, KE + PE should equal E_total. If calculations don't satisfy this, check for errors in height or velocity calculations.
Pendulum Types Comparison
| Pendulum Type | Period Formula | Energy Conservation | Applications |
|---|---|---|---|
| Simple Pendulum | T = 2π√(L/g) | ✅ Yes (ideal) | ✅ Clocks, demos |
| Physical Pendulum | T = 2π√(I/(mgd)) | ✅ Yes (ideal) | ✅ Real objects |
| Damped Pendulum | T ≈ 2π√(L/g) | ❌ No (energy loss) | ✅ Real systems |
| Conical Pendulum | T = 2π√(L cos θ/g) | ✅ Yes (ideal) | ✅ Circular motion |
Frequently Asked Questions
Q: Does mass affect pendulum energy?
A: Yes for total energy (E = mgh_max), but no for maximum velocity! v_max = √(2gh) is independent of mass. A heavier pendulum has more energy but the same speed at the bottom. This demonstrates that kinetic energy depends on velocity squared, not mass directly.
Q: Why does a real pendulum eventually stop?
A: Energy is lost to air resistance and friction at the pivot. The mechanical energy gradually converts to thermal energy through these dissipative forces. Our ideal calculations assume no energy loss, but real pendulums follow exponential decay: E(t) = E₀e^(-γt).
Q: Where is potential energy measured from?
A: We typically set PE = 0 at the lowest point of swing. This is arbitrary—only changes in PE matter for physics. You could also measure from the floor or ceiling, but the lowest point is most convenient for pendulum calculations.
Q: How does amplitude affect pendulum period?
A: For small angles (< 15°), period is independent of amplitude. For larger angles, period increases slightly. The exact formula uses elliptic integrals: T = 4√(L/g)K(sin(θ_max/2)), where K is the complete elliptic integral of the first kind.
Q: What is the relationship between kinetic and potential energy?
A: They are complementary and sum to constant total energy: KE + PE = E_total. When KE is maximum (at bottom), PE is zero. When PE is maximum (at extremes), KE is zero. At any intermediate point, KE = E_total - PE = mgh_max - mgh.
Q: Can a pendulum have negative kinetic energy?
A: No, kinetic energy is always non-negative since KE = ½mv² and velocity squared is always positive. Kinetic energy represents energy of motion, so it cannot be negative. If calculated KE is negative, check for sign errors in velocity calculations.
Q: How do I calculate velocity at any point in the swing?
A: Use energy conservation: v = √(2gL(cos θ - cos θ_max)), where θ is current angle and θ_max is maximum angle. At the bottom (θ = 0), this simplifies to v_max = √(2gL(1 - cos θ_max)) = √(2gh_max).
Pendulum Energy by the Numbers
Official Data Sources
NIST Physical Measurement Laboratory
National Institute of Standards and Technology - Physical constants and measurement standards
MIT OpenCourseWare - Classical Mechanics
MIT OpenCourseWare classical mechanics covering pendulum dynamics and energy conservation
Physics Hypertextbook - Pendulum
Comprehensive pendulum physics, energy transformations, and harmonic motion principles
HyperPhysics - Pendulum Motion
HyperPhysics comprehensive pendulum physics and energy conservation principles
Last Updated: February 7, 2026
⚠️ Disclaimer
This calculator provides estimates based on ideal pendulum physics assuming no friction, air resistance, or energy losses. Results should be verified for real-world applications where damping effects are significant. For large angles (> 15°), the small-angle approximation may introduce errors. Always consider safety factors for engineering applications involving pendulums.
Real-World Applications
Timekeeping
- • Grandfather clocks
- • Metronomes
- • Historical precision timing
Physics Demonstrations
- • Foucault pendulum (Earth rotation)
- • Energy conservation demos
- • Bowling ball pendulum
Engineering
- • Wrecking balls
- • Amusement park rides
- • Seismometers
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