Calculate kinetic and potential energy of a simple pendulum at any angle. Energy conservation, velocity, and height analysis.
The **Pendulum Kinetic Energy Calculator** analyzes the energy transformations in a simple pendulum as it swings back and forth. At any given point in its arc, the pendulum has both kinetic energy (from its motion) and potential energy (from its height above the lowest point). The fundamental principle of energy conservation tells us that the total mechanical energy remains constant throughout the swing, assuming no friction or air resistance.
When a pendulum is released from its maximum angle, all its energy is potential energy (PE = mgh). As it swings downward, potential energy converts to kinetic energy (KE = ½mv²), reaching maximum kinetic energy and velocity at the bottom of the arc. This continuous exchange between kinetic and potential energy is one of the most elegant demonstrations of energy conservation in classical mechanics.
This calculator lets you specify the pendulum length, bob mass, release angle, and any intermediate angle to see the exact energy split at that position. It computes the velocity, height, kinetic energy, and potential energy, and displays a complete breakdown table showing how energy transforms throughout the swing. Whether you're studying physics, designing a pendulum clock, or analyzing oscillation dynamics, this tool provides the detailed energy analysis you need.
Understanding pendulum energy is essential for physics students studying conservation of energy, mechanical oscillations, and harmonic motion. This calculator goes beyond simple formulas by providing a complete energy profile at any angle, letting you visualize how kinetic and potential energy trade off throughout the swing.
Engineers and clockmakers use pendulum energy analysis to design precise timekeeping mechanisms, seismometers, and vibration dampeners. The energy breakdown table is especially useful for identifying where the pendulum moves fastest and where forces are greatest, which matters for structural design and dynamic analysis.
Kinetic Energy: KE = ½mv² Potential Energy: PE = mgh where h = L(1 − cos θ) Total Energy: E = mgh₀ = mgL(1 − cos θ₀) Velocity at angle θ: v = √(2gL(cos θ − cos θ₀)) Variables: m = mass, g = gravitational acceleration, L = length, θ = current angle, θ₀ = release angle
Result: 2.63 J kinetic energy at bottom
A 1 m pendulum with a 2 kg bob released from 30° has maximum height h = 1×(1−cos30°) = 0.134 m. Total energy = 2×9.807×0.134 = 2.63 J. At the bottom (0°), all energy is kinetic, so KE = 2.63 J and velocity = √(2×2.63/2) = 1.62 m/s.
The simple pendulum is one of the most fundamental demonstrations of mechanical energy conservation. As the pendulum swings, energy continuously converts between two forms: gravitational potential energy (proportional to height) and kinetic energy (proportional to the square of velocity). At every point in the arc, the sum of these two energies equals the total mechanical energy, which was set by the initial release conditions.
The mathematics behind this is elegant. At release angle θ₀ from vertical, the bob sits at height h₀ = L(1−cosθ₀) above the lowest point. As it swings down to angle θ, it drops to height h = L(1−cosθ). The potential energy lost (mgh₀ − mgh) converts entirely to kinetic energy, giving the bob velocity v = √(2gL(cosθ − cosθ₀)).
Pendulum energy analysis has numerous real-world applications. In horology (clockmaking), understanding the energy stored in a pendulum swing is crucial for designing escapement mechanisms that maintain amplitude with minimal energy input. The total energy determines how much driving force the clock mechanism must supply.
In seismology, pendulum-based instruments use the energy of ground motion to detect and measure earthquakes. The conversion between kinetic and potential energy in the seismometer's pendulum faithfully records ground acceleration. Similarly, Foucault pendulums in science museums are carefully designed with specific mass, length, and initial energy to demonstrate Earth's rotation over many hours.
This calculator assumes a point-mass bob on a massless string — the ideal simple pendulum. Real pendulums have distributed mass (physical pendulums), air resistance, and pivot friction. For the physical pendulum case, see our Physical Pendulum Calculator, which accounts for the moment of inertia of extended bodies. For energy analysis precision, note that friction effects become significant only after many oscillation cycles, so the ideal model works well for analyzing individual swings.
No. The velocity at the bottom depends only on the release angle and pendulum length: v = √(2gL(1−cosθ₀)). Mass cancels out in the energy conservation equation. However, heavier pendulums have more kinetic energy at the same velocity.
This calculator models an ideal simple pendulum for physics analysis. In real pendulums, air resistance and pivot friction slowly drain energy, causing the amplitude to decrease over time. For short-term analysis, the ideal model is very accurate.
The calculator handles angles up to 180°. At 90°, the bob is level with the pivot. Beyond 90°, the bob is above the pivot. At exactly 180° (inverted), all energy is potential and the position is unstable.
Total energy equals the potential energy at the release angle: E = mgL(1−cosθ₀). This is the maximum PE, which equals the maximum KE at the bottom. Energy is conserved throughout the swing.
Yes! Change the gravitational acceleration to the Moon value (1.625 m/s²). The pendulum will have less total energy and swing more slowly, but the energy conservation principle still applies identically.
Velocity at any angle is v = √(2gL(cosθ − cosθ₀)). It is maximum at θ=0 and zero at θ=θ₀. The relationship is nonlinear — the pendulum accelerates fastest near the bottom of its swing.