Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_163

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The figure shows the end view of a long homogeneous solid cylinder which floats in a liquid and has a removed segment. Show that and are the two values of the angle between its centerline and the vertical for which the cylinder floats in stable positions. Problem 5/163 (a) Analysis of Equilibrium and Stability 1. Centroid Location: Due to the homogeneity of the cylinder and the symmetry of its cross-section, the center of gravity must lie on the axis of symmetry, which is identified as the centerline in the problem diagram. 2. Buoyancy and Center of Buoyancy: For a floating body, the buoyant force acts at the center of buoyancy , which is the centroid of the submerged volume. In this 2D cross-sectional view, is the centroid of the submerged area. 3. Geometry of the Submerged Part: Assuming the cylinder floats such that only its circular boundary is in contact with the liquid (or that the submerged part is a segment of the original circle), the center of buoyancy will always lie on the vertical line passing through the center of the original circle. This is because any circular segment submerged is symmetric about the vertical diameter of the circle. 4. Equilibrium Condition: For a floating body to be in static equilibrium, the center of gravity and the center of buoyancy must lie on the same vertical line. Since the center of the circle and the center of buoyancy are always on a vertical line, equilibrium requires that also be on