. (The tautochrone) The parametric equations x D a. ? sin /
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. (The tautochrone) The parametric equations x D a. ? sin / and y D a.cos ? 1/ (for 0 2 ), describe an arch of the cyclo
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. (The tautochrone) The parametric equations x D a. ? sin / and y D a.cos ? 1/ (for 0 2 ), describe an arch of the cycloid followed by a point on a circle of radius a rolling along the underside of the x-axis. Suppose the curve is made of wire along which a bead can slide without friction. (See Figure 11.30.) If the bead slides from rest under gravity, starting at a point having parameter value 0, show that the time it takes for the bead to fall to the lowest point on the arch (corresponding to D ) is a constant, independent of the starting position 0. Thus, two such beads released simultaneously from different positions along the wire will always collide at the lowest point. For this reason, the cycloid is sometimes called the tautochrone, from the Greek for
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