✓ Sources checked · 10 September 2026 · View verification
An inverted cycloid is a tautochrone: in uniform gravity, ideal beads released from rest at different points on one side reach its lowest point in the same time.
By Matan · Updated
The endpoint matters
The equal-time statement concerns release from rest and arrival at the common bottom, not travel between arbitrary pairs of points. For generating-circle radius a, the descent time is π√(a/g). Friction and rolling inertia are excluded in the sliding-particle model. [1]
Why different heights can take equal time
Let s be signed arc length from the bottom. Cycloid geometry gives height above the bottom y = s²/(8a). Gravitational potential is therefore mgs²/(8a). The tangential equation is s̈ = −gs/(4a), a harmonic oscillator with angular frequency √(g/(4a)). From any nonzero release position at rest, reaching s = 0 takes a quarter cycle, π√(a/g).
Worked equal-time descent
With a = 0.25 m and g = 9.8 m/s², the descent time is π√(0.25/9.8) = 0.502 s. A higher release gives more distance but also greater speed. Starting already at the bottom is the trivial zero-distance case and is not a distinct timed descent.
Equal time and fastest time are different questions
The brachistochrone asks for the fastest path between specified endpoints. Its classical solution is also a cycloid, but the optimization problem is different. The tautochrone property does not establish universal designs for spillways, satellite orbits or roller coasters.
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How this article was checked
Sources checked; calculations checked where shown. Main claims and model assumptions were checked against the references below. Example arithmetic and units were checked separately. Numerical examples are illustrative calculations, not experimental measurements.
Source checking and editing assisted by AI. No independent human scientific review is recorded for this revision. This record applies to the version dated 10 September 2026.
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Cite this article
Matan. “Tautochrone curve.” Your Physicist. Originally published 2023-03-17; updated 2026-09-10. https://your-physicist.com/tautochrone-curve/
Revision history
10 September 2026: Corrected the arbitrary-endpoint definition and unsupported applications; derived the equal descent time. Added references and a transparent verification record.
