Concept explainers
The Tautochrone. A problem of interest in the history of mathematics is that of finding the Tautochrone-the curve down which a particle will slide freely under gravity alone, reaching the bottom in the same time regardless of its starting point on the curve. This problem arose in the construction of a clock pendulum whose period is independent of the amplitude of its motion. The Tautochrone was found by Christian Huygens in
The geometric configuration is shown in Figure
Then it follows from the principle of conservation of energy that the time
(a) Assume that
Then show that
Hint: See Problem 37 of Section 5.1.
(b) Combining Eqs. (i) and (iv), show that
where
(c) Use the substitution
Equations (vi) can be identified as parametric equations of a cycloid. Thus the Tautochrone is an arc of a cycloid.
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Differential Equations: An Introduction to Modern Methods and Applications
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