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- Using integration, locate the centroid of the area under the n-th order parabola in terms of b, h, and n (n is a positive integer). (b) Check the result of part (a) with Table 8.1 for the case n = 2.The 12-ft wide quarter-circular gate AB is hinged at A. Determine the contact force between the gate and the smooth surface at B due to water pressure acting on the gate. Use =62.4lb/ft3 for water.Compute the surface area of the axi-symmetric domed structure.
- Determine the smallest distance I) that would prevent the concrete dam from tipping about corner A. Use w=62.4lb/ft3 for water and c=150lb/ft3 for concrete.Use integration to locate the centroid of the shaded region in terms of R and t. (b) Show that when t0 the result of part (a) agrees with that given in Table 8.2 for a quarter circular arc.One side of the container has a 03-m square door that is hinged at its top edge. If the container is filled with water, determine the smallest force F that must be applied to the bottom edge of the door to keep it closed.
- Use numerical integration to locate the centroid of the symmetric planeUsing the method of composite areas, find the dimension h that maximizes the centroid coordinate y of the plane region shown.Use integration to locate the centroid of the volume of the hemisphere. Compare your results with Table 8.3.
- Using the method of composite areas, find the centroid of the truncated parabolic complement.The cylindrical water tank with R = 10 ft and H = 1.6 ft has thin steel walls of uniform thickness and weighs 18000 lb when empty. Determine the depth of the water h for which the center of gravity G of the tank plus water will be located at the surface of the water. For water, use =62.4lb/ft3.The cylindrical container will have maximum stability against tipping when its centroid is located at its lowest possible position. Determine the depth h of the cylindrical portion that must be removed to achieve this.