The Tautochrone. A problem of interest in the history of mathematics is that of

Chapter 6, Problem 29

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The Tautochrone. A problem of interest in the history of mathematics is that of findingthe tautochrone6the 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. Thisproblem arose in the construction of a clock pendulum whose period is independent ofthe amplitude of its motion. The tautochrone was found by Christian Huygens (16291695) in 1673 by geometrical methods, and later by Leibniz and Jakob Bernoulli usinganalytical arguments. Bernoullis solution (in 1690) was one of the first occasions on whicha differential equation was explicitly solved. The geometric configuration is shown inFigure 6.6.2. The starting point P(a, b) is joined to the terminal point (0, 0) by the arc C.Arc length s is measured from the origin, and f(y) denotes the rate of change of s withrespect to y:f(y) = dsdy =1 +dxdy 21/2. (i)Then it follows from the principle of conservation of energy that the time T(b) requiredfor a particle to slide from P to the origin isT(b) = 12g b0f(y)b ydy. (ii)y(a) Assume that T(b) = T0, a constant, for each b. By taking the Laplace transform ofEq. (ii) in this case, and using the convolution theorem, show thatF(s) =2gT0 s; (iii)then show thatf(y) =2gT0 y. (iv)Hint: See of Section 6.1.(b) Combining Eqs. (i) and (iv), show thatdxdy =2 yy, (v)where = gT20 /2.(c) Use the substitution y = 2 sin2(/2) to solve Eq. (v), and show thatx = ( + sin ), y = (1 cos ). (vi)Equations (vi) can be identified as parametric equations of a cycloid.Thus the tautochroneis an arc of a cycloid.

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