Question

MAE 242. Introduction to Fluid Mechanics. Homework Assignment 3. Due October 2. 1. A block of solid polyethylene (o=860 kg/mº) with triangular cross section is floating in water. The cross section is an isosceles triangle with angles x/2, /4, A/4 as shown. (a) For orientation a, find the depth of the block. Is it possible for this orientation to be unstable? Explain. (b) For orientation b, find the depth of the block Is it possible for this orientation to be unstable? Explain 10 cm 10 cm Figure for problem 1.

Answer 1

Write the given data as follow: Density of solid, P: = 860 kg/m? Angles of isosceles triangle are: 1/2, 1/4, and 1/4 Draw the cross-sectional view for orientation (a). 5 X 5 2 Find the value of y. 5 5 2 X 2 y = 5 - 2x For equilibrium gravity force will be balanced by buoyant force. Pw*Vn* g = P. XV.xg 1000 X 5 -X y X -X = 860 x x 5 2 2 (5-2x)x(5 – 2x) 25 1000 ($ 2.) |- 60. 4 4 1000[-4x² + 20x ] = 860x 25 860 -4x2 + 20x = 40 4x2 – 20x +21.5 = 0 Solving the equation, we get: x = 1.5645 cm As the Metacenter is above the Center of mass therefore the orientation is not unstable.

Draw the cross-sectional view for orientation (b). 5 5 2 2 5 x 2 x Find the value of y. 5 = 5 2 y . y = 2x For equilibrium gravity force will be balanced by buoyant force. PxVn$ = P. XV. X 8 1 5 1000 -X y = 860 x– x 5x 2 2 1000 x x 2.x x x = 860 x X 25 1000 x 2x2 = 860 x 2 .x = 2.3184 cm As the Metacenter is below the Center of mass therefore the orientation is unstable.