Pendulum Energy at a Point Calculator
Split a swinging pendulum’s energy into potential and kinetic at any angle, and find its speed there.
How it works
height h = L·(1 − cosθ) · total E = m·g·L·(1 − cosθ₀) · PE = m·g·L·(1 − cosθ) · KE = E − PE · v = √(2g·L·(cosθ − cosθ₀))
A frictionless pendulum conserves its mechanical energy, trading potential for kinetic as it swings. At the release angle it is momentarily at rest, so all the energy is potential; at the lowest point that energy has become entirely kinetic, giving the maximum speed. The height of the bob above the lowest point is L(1 − cosθ), and the speed anywhere follows from √(2gL(cosθ − cosθ₀)) — notice the mass cancels, so speed does not depend on how heavy the bob is.
Worked example
A 0.5 kg bob on a 1 m string released from 30° has total energy m·g·L·(1 − cos30°) = 0.5 × 9.81 × 1 × 0.134 ≈ 0.657 J. At the lowest point (0°) all of it is kinetic, so v = √(2 × 9.81 × 1 × (1 − cos30°)) ≈ 1.62 m/s.
Sources: Pendulum (mechanics) — Wikipedia
Frequently asked questions
Where is the pendulum fastest?
At the lowest point of the swing, where all the potential energy from the release height has converted to kinetic energy. There the speed is √(2gL(1 − cosθ₀)).
Does a heavier bob swing faster?
No. The mass cancels out of the energy equation, so the speed at any point depends only on the length, the angles and gravity — not on how heavy the bob is. A heavy and a light bob released from the same angle move identically (ignoring air resistance).
Why can’t the point angle be larger than the release angle?
Energy conservation means the bob can never rise higher than the point it was released from. Asking for a larger angle would require kinetic energy to be negative, which is impossible, so the calculator flags it.
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This calculator is for educational purposes. Double-check important results before acting on them.