A circular helix that lies on the cylinder x2 + y2 = R2

Chapter , Problem 39

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A circular helix that lies on the cylinder x2 + y2 = R2 with pitch may be described parametrically by x = R cos , y = R sin , z = , 0. A particle slides under the action of gravity (which acts parallel to the z axis) without friction along the helix. If the particle starts out at the height z0 > 0, then when it reaches the height z along the helix, its speed is given by ds dt = (z0 z)2g, where s is arc length along the helix, g is the constant of gravity, t is time, and 0 z z0. (a) Find the length of the part of the helix between the planes z = z0 and z = z1, 0 z1 < z0. (b) Compute the time T0 it takes the particle to reach the plane z = 0.