About the Kinetic Energy of a Pendulum
A simple pendulum — a point mass on a massless string or rod of length L — continuously exchanges energy between kinetic and potential as it swings. Released from rest at a maximum angle θ₀ from vertical, conservation of mechanical energy gives the pendulum's kinetic energy at any angle θ (with |θ| no greater than θ₀) as KE = mgL(cosθ − cosθ₀), where m is the bob's mass and g is the local gravitational acceleration. This is the exact energy-conservation result, not the small-angle approximation used to estimate the pendulum's period — it holds for any swing amplitude up to (but not including) 180°.
Key principles behind the calculation
- Conservation of energy: in an ideal, frictionless swing, kinetic energy (KE) plus potential energy (PE) is constant: KE + PE = E = mgL(1 − cosθ₀), the total mechanical energy fixed by the release angle.
- Height drop from the release point: at angle θ the bob has fallen a height h = L(cosθ − cosθ₀) below where it was released, so KE = mgh follows directly from the work-energy theorem.
- Speed from kinetic energy: once KE is known, the bob's instantaneous speed follows from KE = ½mv², so v = √(2·KE / m); this is largest at θ = 0 (the bottom of the swing) and zero at the turning points θ = ±θ₀.
Assumptions and getting accurate results
- The formula assumes an idealized simple pendulum: a point mass on a massless, inextensible string or rigid rod, swinging in one vertical plane with no air resistance or pivot friction.
- The current angle θ can never exceed the release amplitude θ₀ in magnitude — in an ideal system the bob never swings higher than the point it started from.
- This calculator converts your chosen length unit to meters internally and reports energy in joules and speed in meters per second, using whatever value of g you enter (9.81 m/s² for Earth by default).
- Real pendulums lose a small amount of energy to air resistance and friction each swing, so a measured speed at the bottom of the swing will be slightly lower than this ideal calculation predicts.
Frequently Asked Questions
What is the formula for the kinetic energy of a pendulum?
At any angle θ from vertical (with |θ| no greater than the release amplitude θ₀), the kinetic energy is KE = mgL(cosθ − cosθ₀), where m is the bob's mass, L is the pendulum's length, and g is gravitational acceleration. This comes directly from conservation of energy: whatever height the bob has fallen since release converts into kinetic energy.
Where is a pendulum's kinetic energy greatest?
Kinetic energy is maximum at the bottom of the swing (θ = 0), where it equals the full mechanical energy of the system: KE_max = mgL(1 − cosθ₀). At that point the bob's speed is also at its maximum, v_max = √(2gL(1 − cosθ₀)). Kinetic energy falls to zero at the two turning points, θ = +θ₀ and θ = −θ₀, where the bob is momentarily at rest.
Does this calculator use the small-angle approximation?
No. The formula KE = mgL(cosθ − cosθ₀) comes from exact conservation of energy and is valid for any swing amplitude up to (but not including) 180°. That differs from the small-angle period formula T = 2π√(L/g), which only stays accurate for amplitudes below roughly 15°.