Question

6.14 ** (a) Prove that the brachistochrone curve (6.26) is indeed a cycloid, that is, the curve traced by a point on the circumference of a wheel of radius a rolling along the underside of the x axis (b) Although the cycloid repeats itself indefinitely in a succession of loops, only one loop is relevant to the brachistochrone problem. Sketch a single loop for three different values of a (all with the same starting point 1) and convince yourself that for any point 2 (with positive coordinates x2, y2) there is exactly one value of a for which the loop goes through the point 2. (c) To find the value of a for a given point x2, y2 usually requires solution of a transcendental equation. Here are two cases where you can do it more simply: For x2 = π b, y-2b and again for x,-2π b. У2-0 find the value of for which the cycloid goes through the point 2 and find the corresponding minimum times. W1

Answer 1

Given That: - The curve traced by a point on The Circumference of a wheel of radius 'a' along The x - axis. We take The line to be the x - axis. Make a point p on the wheel. We start The Wheel with P at origin and volt The wheel to The right. We use The angle t Through which The wheel Turns. The base of wheel lies at units trans The origin. The wheels center C lies at (at, a) cos Theta = x - at/a a cos Theta = x - at \r\n x = at + a cos Theta Sin Theta = y - a/a y - a = a sin Theta y = a + a sin Theta from diagram t + Theta = 3pi/2 Theta = 3pi/2 - t x = at + a cos (3pi/2 - t) x = at - a sin t y = a + a sin (3 pi/2 - t) y = a - a cos t