Everything about Tautochrone Curve totally explained
A
tautochrone or
isochrone curve is the curve for which the time taken by an object sliding without friction in uniform
gravity to its lowest point is independent of its starting point. The curve is a
cycloid, and the time is equal to
π times the
square root of the radius over the acceleration of gravity.
The tautochrone problem
The tautochrone problem, the attempt to identify this curve, was solved by
Christiaan Huygens in
1659. He proved geometrically in his
Horologium oscillatorium (
The Pendulum Clock,
1673) that the curve was a
cycloid. This solution was later used to attack the problem of the
brachistochrone curve.
Jakob Bernoulli solved the problem using calculus in a paper (
Acta Eruditorum,
1690) that saw the first published use of the term
integral.
Later mathematicians such as
Joseph Louis Lagrange and
Leonhard Euler looked for an analytical solution to the problem.
"Virtual gravity" solution
Perhaps the simplest solution to the tautochrone problem is to note a direct relation between the angle of an incline and the gravity felt by a particle on the incline. A particle on a 90° vertical incline feels the full effect of gravity, while a particle on a horizontal plane feels effectively no gravity. At intermediate angles, the "virtual gravity" felt by the particle is
. The first step is to find a "virtual gravity" that produces the desired behavior.
The "virtual gravity" required for the tautochrone is simply proportional to the distance remaining to be traveled, which admits a simple solution:
»
It can be shown that the cycloid obeys this equation.
(
Simmons, Section 54).
Further Information
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