Last Updated: October 30, 2025
Calculate the kinetic energy of a pendulum from mass and speed using KE = 1/2 m v² or from drop height using energy conservation KE ≈ m g h. Essential tool for physics energy analysis, oscillatory motion, and understanding energy transformations in pendulum systems. Perfect for students, educators, and engineers analyzing pendulum dynamics, energy conservation, and mechanical oscillations.
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The Kinetic Energy of a Pendulum Calculator is a specialized physics tool that calculates the kinetic energy of a pendulum bob at various points in its motion. A pendulum continuously exchanges energy between gravitational potential energy (at highest points) and kinetic energy (at the lowest point), providing an excellent demonstration of energy conservation.
This calculator is essential for analyzing pendulum dynamics, understanding energy conservation, physics education, and experimental measurements. It provides accurate calculations using two methods: from measured speed (KE = (1/2)mv²) or from drop height (KE ≈ mgh), enabling students and researchers to understand energy transformations in oscillatory systems.
Kinetic Energy (KE): Energy of motion, measured in joules (J). For a pendulum bob, KE = (1/2)mv², where m is mass and v is speed. Maximum kinetic energy occurs at the lowest point of the swing, where speed is maximum and potential energy is minimum. Kinetic energy is zero at the highest points where the bob momentarily stops.
Potential Energy (PE): Gravitational energy due to height, measured in joules (J). PE = mgh, where m is mass, g is gravitational acceleration, and h is height above the lowest point. Maximum potential energy occurs at the highest points of the swing. As the bob falls, potential energy converts to kinetic energy.
Energy Conservation: In an ideal lossless pendulum, total mechanical energy (KE + PE) remains constant. Energy continuously converts between kinetic and potential forms. At any point: KE + PE = constant. This conservation principle allows calculating KE from PE changes or vice versa.
Mass (m): Mass of the pendulum bob, measured in kilograms (kg). Kinetic energy is directly proportional to mass—doubling mass doubles KE for the same speed. Mass affects both potential and kinetic energy equally, canceling out in energy ratio calculations.
Speed (v): Linear velocity of the bob, measured in meters per second (m/s). Kinetic energy depends quadratically on speed (KE ∝ v²), meaning doubling speed quadruples kinetic energy. Speed varies throughout the swing, maximum at the bottom, zero at the extremes.
Drop Height (h): Vertical distance the bob falls from release point to evaluation point, measured in meters (m). For small angles, h ≈ L(1 - cos θ₀), where L is length and θ₀ is release angle. Energy conservation gives KE ≈ mgh at the lowest point, assuming negligible losses.
A pendulum demonstrates continuous energy transformation. When released from rest at a height, all energy is potential. As the bob falls, potential energy converts to kinetic energy, reaching maximum kinetic energy at the lowest point where speed is maximum. The bob then climbs, converting kinetic energy back to potential energy. This cycle repeats, with energy oscillating between the two forms.
The two calculation methods reflect this energy conservation. The speed-based method (KE = (1/2)mv²) directly measures the actual kinetic energy when speed is known. The height-based method (KE ≈ mgh) estimates kinetic energy from the potential energy lost, assuming perfect energy conversion. In real systems with losses (air drag, friction), the height-based estimate provides an upper bound—actual KE will be slightly less due to dissipative losses.
The energy exchange drives pendulum motion. Maximum kinetic energy at the bottom provides the momentum needed to climb the opposite side. The continuous conversion between KE and PE sustains oscillation. Understanding this energy flow explains why pendulums oscillate and how amplitude relates to initial energy. Energy conservation is fundamental to pendulum behavior, whether analyzing simple pendulums, physical pendulums, or complex oscillatory systems.
The study of pendulums has shaped understanding of mechanics for centuries. Galileo Galilei observed the near-isochronous behavior of pendulums, leading to pendulum clocks. The energy conservation principle was developed as part of classical mechanics, with pendulums serving as transparent demonstrations of energy transformations. Understanding pendulum kinetic energy became crucial for timekeeping, experimental physics, and teaching energy conservation principles.
Energy: J (Joule) - same for kinetic and potential energy
Mass: kg (kilograms)
Speed: m/s (meters per second)
Height: m (meters) - vertical distance
Gravitational Acceleration: m/s² - Earth: 9.81 m/s²
Relationships: KE = (1/2)mv², KE ≈ mgh (at lowest point in ideal system)
KE = (1/2)mv²
Where:
Use case: Calculate kinetic energy directly from measured or known speed. This is the fundamental kinetic energy formula, valid at any point in the pendulum's motion. Use when speed can be measured (photogates, motion tracking, video analysis) or calculated. Provides the actual kinetic energy, accounting for any energy losses. Essential for experimental measurements and precise calculations.
KE ≈ mgh
Where:
Use case: Estimate kinetic energy from potential energy conservation. Valid at the lowest point in an ideal (lossless) system. Uses energy conservation: the potential energy lost equals kinetic energy gained. Convenient when height is easier to measure than speed. In real systems with losses, this provides an upper bound—actual KE will be slightly less. Useful for quick estimates and energy conservation demonstrations.
KE ≈ mgL(1 - cos θ₀)
Where:
Use case: Calculate kinetic energy from pendulum geometry when length and release angle are known. The height h = L(1 - cos θ₀) represents the vertical drop from release to lowest point. For small angles, cos θ₀ ≈ 1 - θ₀²/2, giving h ≈ Lθ₀²/2. Useful when angle and length are known but direct height measurement is difficult. Convenient for pendulum experiments and calculations.
KE = (1/2)m(ωL)²
Where:
Use case: Calculate kinetic energy from angular velocity, using v = ωL to convert to linear speed. Useful when angular velocity is measured (optical encoders, high-speed cameras) or calculated. Convenient for pendulum analysis where angular motion is tracked. The relationship v = ωL connects angular and linear motion for pendulum calculations.
Pendulum kinetic energy calculations are essential across numerous physics, engineering, and educational fields. Here's a comprehensive overview of practical applications:
| Industry | Applications | Importance |
|---|---|---|
| Physics Education | Energy conservation demonstrations, kinetic energy experiments, pendulum labs, energy transformation studies, mechanics teaching | Fundamental for teaching energy conservation and mechanics |
| Timekeeping & Metrology | Pendulum clock design, precision timekeeping, gravitational acceleration measurement, metrology applications | Important for historical and modern timing systems |
| Seismology | Seismometer design, ground motion detection, earthquake monitoring, vibration sensors, seismic instruments | Critical for earthquake detection and monitoring |
| Experimental Physics | Energy conservation experiments, damping studies, oscillation analysis, measurement techniques, laboratory exercises | Essential for experimental physics and research |
| Mechanical Engineering | Vibration analysis, swinging systems, energy absorption, motion control, oscillatory systems | Important for mechanical system design |
| Entertainment & Design | Pendulum clocks, kinetic sculptures, amusement rides, decorative pendulums, artistic installations | Useful for functional and aesthetic applications |
Given:
Step 1: Apply kinetic energy formula
KE = 1/2 m v²
KE = 0.5 × 2.0 × (3.0)²
KE = 0.5 × 2.0 × 9 = 9.0 J
Final Answer
9.0 J
Given:
Step 1: Apply height-based energy estimate
KE ≈ m g h
KE ≈ 1.5 × 9.80665 × 0.25
KE ≈ 3.68 J
Final Answer
3.68 J
Given:
Step 1: Compute vertical drop
h = L(1 - cos θ₀) = 0.75(1 - cos 12°) ≈ 0.0169 m
Step 2: Estimate kinetic energy
KE ≈ m g h = 0.60 × 9.80665 × 0.0169 ≈ 0.099 J
Final Answer
≈ 0.099 J
Given:
Height method:
KEₕ ≈ m g h = 0.80 × 9.80665 × 0.05 ≈ 0.392 J
Speed method:
KEᵥ = 1/2 m v² = 0.5 × 0.80 × (0.95)² ≈ 0.361 J
Difference indicates ~0.031 J of losses (air drag, friction).
Final Answers
KEₕ ≈ 0.392 J; KEᵥ ≈ 0.361 J
Given:
Step 1: Convert to linear speed
v = ω L = 1.6 × 0.90 = 1.44 m/s
Step 2: Apply KE formula
KE = 1/2 m v² = 0.5 × 1.2 × (1.44)² ≈ 1.244 J
Final Answer
≈ 1.244 J
Given:
Step 1: Small-angle height
h ≈ L(1 - cos θ₀) ≈ 1.20(1 - cos 6°) ≈ 0.0066 m
Step 2: Estimate KE
KE ≈ m g h = 0.40 × 9.80665 × 0.0066 ≈ 0.026 J
Final Answer
≈ 0.026 J
Disclaimer: The calculators and tools are for educational purposes. Verify results independently before professional use.