Pendulum Kinetic Energy Calculator
Find the maximum kinetic energy and speed of a pendulum bob at the lowest point of its swing using conservation of energy from the release angle.
kg
m
°
m/s²
KE_max = mgL(1 − cosθ) — equals the potential energy at release
- 1
cos(θ)
cos(20°) = 0.939693 - 2
Height above equilibrium h = L × (1 − cosθ)
1 × (1 − 0.939693) = 0.0603 m - 3
Max kinetic energy KE = m × g × h
0.5 × 9.81 × 0.0603 = 0.2958All potential energy at the release angle converts to kinetic energy at the bottom.
How does this calculator work?
Release a pendulum bob from angle θ and it reaches maximum kinetic energy KE = mgL(1 − cosθ) at the bottom. Maximum speed is v = √(2gL(1 − cosθ)). Mass cancels from the speed but not from the energy. Valid for any angle 0–89° in a frictionless model.
Formula
How this is calculated
When a pendulum bob is released from rest at angle θ from the vertical, its centre-of-mass height above the equilibrium (lowest) point is h = L(1 − cosθ), where L is the string length. At the moment of release the bob has gravitational potential energy PE = mgh and zero kinetic energy. As it swings down, potential energy converts to kinetic energy; at the lowest point all PE has been converted, giving maximum kinetic energy KE_max = mgh = mgL(1 − cosθ). From KE = ½mv² the maximum speed is v_max = √(2gL(1 − cosθ)).
Conservation of mechanical energy assumes no friction, air resistance, or string elasticity. The energy stored at the top is exactly the kinetic energy at the bottom, so mass cancels from the speed formula — but not from the energy formula (heavier bobs carry more energy at the same speed). The period and frequency shown use the small-angle approximation T = 2π√(L/g).
Angles up to 89° are valid in the energy formula (which is exact for the conservative frictionless model). However, the small-angle period approximation loses accuracy beyond about 15°; see the Pendulum Period Calculator for the large-angle correction.
Frequently asked questions
At the lowest point the pendulum bob has its minimum potential energy, so all the energy released as it fell from the release angle has been converted to kinetic energy. By conservation of energy, KE is maximum where PE is minimum.
No — mass cancels from v_max = √(2gL(1 − cosθ)), so two bobs of different masses on the same string released from the same angle reach the same speed. However, the heavier bob has more kinetic energy (KE = ½mv²).
Friction and air resistance dissipate mechanical energy as heat; each successive swing is slightly smaller (damped oscillation). This calculator models the ideal frictionless case, so KE_max = initial PE with no losses.
Also known as
TG we-Calculate Editorial Team. (2026). Pendulum Kinetic Energy Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pendulum-kinectic-energy-calculator
TG we-Calculate Editorial Team. "Pendulum Kinetic Energy Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/pendulum-kinectic-energy-calculator.
TG we-Calculate Editorial Team, "Pendulum Kinetic Energy Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pendulum-kinectic-energy-calculator
@misc{wecalculate_pendulum_kinectic_energy_calculator, title = {Pendulum Kinetic Energy Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pendulum-kinectic-energy-calculator}}, year = {2026}, note = {TG we-Calculate} }
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