Circular Motion and Gravitation

['r = R sinθ (where r is the radius of the circle of rotation for a point P, R is the radius of the Earth, and θ is the colatitude)', "g' = √(g² - ω²R sin²θ(2g - ω²R)) (where g' is the effective acceleration due to gravity, g is the true acceleration due to gravity, ω is the angular speed of the Earth, R is the radius of the Earth, and θ is the colatitude)", 'tan α = (ω²R sinθ cosθ) / (g - ω²R sin²θ) (where α is the angle between the apparent vertical and the true vertical)']