SECTION 10.7 Plane Curves and Parametric Equations 745 Applications to Mechanics If a 0 < in equation (5), we obtain an inverted cycloid, as shown in Figure 77(a). The inverted cycloid occurs as a result of some remarkable applications in the field of mechanics. We mention two of them: the brachistochrone and the tautochrone . Figure 77 (a) Inverted cycloid A B (b) Curve of quickest descent A B Q (c) All reach Q at the same time NOTE In Greek, brachistochrone means “the shortest time,” and tautochrone “equal time.” j Figure 78 Cycloid Cycloid Cycloid THEOREM Parametric Equations of a Cycloid The parametric equations of a cycloid are x t a t t y t a t sin 1 cos ( ) ( ) ( ) ( ) = − = − (5) The brachistochrone is the curve of quickest descent. If a particle is constrained to follow some path from one point A to a lower point B (not on the same vertical line) and is acted on only by gravity, the time needed to make the descent is least if the path is an inverted cycloid. See Figure 77(b). For example, to slide packages from a loading dock onto a truck, a ramp in the shape of an inverted cycloid might be used, so the packages get to the truck in the least amount of time. This remarkable discovery, which has been attributed to many famous mathematicians (including Johann Bernoulli and Blaise Pascal), was a significant step in creating the branch of mathematics known as the calculus of variations . To define the tautochrone , let Q be the lowest point on an inverted cycloid. If several particles placed at various positions on an inverted cycloid simultaneously begin to slide down the cycloid, they will reach the point Q at the same time, as indicated in Figure 77(c). The tautochrone property of the cycloid was used by Christiaan Huygens (1629–1695), the Dutch mathematician, physicist, and astronomer, to construct a pendulum clock with a bob that swings along a cycloid (see Figure 78). In Huygens’s clock, the bob was made to swing along a cycloid by suspending the bob on a thin wire constrained by two plates shaped like cycloids. In a clock of this design, the period of the pendulum is independent of its amplitude. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.7 Assess Your Understanding 1. The function f x x 3sin 4 ( ) ( ) = has amplitude and period . (pp. 430–432) 2. Suppose x x t( ) = and y y t( ) = are two functions of a third variable t that are defined on the same interval I. The graph of the collection of points defined by x y x t y t , ( , ) ( ) ( ) ( ) = is called a n( ) . The variable t is called a n( ) . 3. Multiple Choice The parametric equations x t t y t t 2 sin 3cos ( ) ( ) = = define a n( ) . (a) circle (b) ellipse (c) hyperbola (d) parabola Concepts and Vocabulary 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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