Packages are moved from point A on the upper floor of a

Chapter 13, Problem 13.78

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Packages are moved from point A on the upper floor of a warehouse to point B on the lower floor, 12 ft directly below A, by means of a chute, the centerline of which is in the shape of a helix of vertical axis y and radius R = 8 ft. The cross section of the chute is to be banked in such a way that each package, after being released at A with no velocity, will slide along the centerline of the chute without ever touching its edges. Neglecting friction, (a) express as a function of the elevation y of a given point P of the centerline the angle f formed by the normal to the surface of the chute at P and the principal normal of the centerline at that point, (b) determine the magnitude and direction of the force exerted by the chute on a 20-lb package as it reaches point B. Hint: The principal normal to the helix at any point P is horizontal and directed toward the y axis, and the radius of curvature of the helix is \(r = R[1 + (h/2pR)^{2}]\).