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Concept
Tautochrone curve
Summary
A tautochrone curve or isochrone curve (from Greek prefixes tauto- meaning same or iso- equal, and chrono time) 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 on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659. He proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve is a cycloid.
On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other ...
The cycloid is given by a point on a circle of radius tracing a curve as the circle rolls along the axis, as:
Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as diameter of the circle that generates the cycloid, multiplied by . In modern terms, this means that the time of descent is , where is the radius of the circle which generates the cycloid, and is the gravity of Earth, or more accurately, the earth's gravitational acceleration.
This solution was later used to solve the problem of the brachistochrone curve. Johann Bernoulli solved the problem in a paper (Acta Eruditorum, 1697).
The tautochrone problem was studied by Huygens more closely when it was realized that a pendulum, which follows a circular path, was not isochronous and thus his pendulum clock would keep different time depending on how far the pendulum swung. After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve.
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