StabilityFloatingCone

StabilityFloatingCone - Stability of a Floating Cone with...

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1 Stability of a Floating Cone with Vertex Down Assume that the cone is right and has height h and that the base is a circle of radius r . Let the angle of the vertex be denoted by ! so that tan 2 ( ) = r h . Assume that the density of the cone is d with 0 < d < 1 with the density of water being 1 . Then for the cone to be at equilibrium in the above position, the depth of the vertex will be at D = 3 h . Now consider the cone slightly tipped at an angle from vertical. Assume that the coordinates are at the vertex of the cone with the y -axis vertical and the x -axis horizontal. The center of gravity of the cone so tipped will have the following x -coordinate. x 2 ( ) = 3 4 h sin( ) The center of gravity of the displaced water will have the following x -coordinate. x 1 ( ) = 3 8 D ( ) tan " 2 + ( ) # tan 2 # ( ) $ % & h r D = d 3 h D ( ) 2 2
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2 This last result was obtained by noting that the displaced water will form a cone with base an ellipse. The length of the major axis of the ellipse will be L = D ( ! ) tan " 2 # ( ) + tan 2 + ( ) $ % & . The centroid of this ellipse will be at the midpoint of the major axis and the centroid of the displaced water will be on the line joining this
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.

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StabilityFloatingCone - Stability of a Floating Cone with...

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