6.7 Fluid Pressure and Fluid Force The pressure at any point on a dam depends only on how Fact #1: far below the water surface the point is and not how much the dam wall happens to be tilted at that point! h (depth) . Pascal's Principle Dams are made thicker at the bottom than at the top Fact #2: because the pressure against them increases with depth! ..... ..... ..... ...... .. ...... .. ...... .. ..... ...... .. ...... .. ...... .. 1 Def. of Fluid Pressure The pressure on an object at depth h in a liquid is Pressure = p = w .h where w is the weight‐density of the liquid per unit of volume. lbs. 62.4 3 water w = /ft p470 list units! p = w .h /lbs. ft/ = ft3/ 2 /lbs. p = ft2 2 The Constant‐Depth (object submerged horizontally ( Formula for Fluid Force: ( Force per . Area of ( Fluid force = F = ( unit area ( object . = pressure Area . = p A F = (w . h ( .A units! 2 / lbs. ft . ft = lbs. Force = ft3 3 Examples: Constant Depth Find the fluid force on a rectangular metal sheet measuring 3 feet by 4 feet that is 1.) submerged horizontally in 6 feet of water. 6 3 4 4 The Variable‐Depth (object submerged vertically ( Integral Formula for Fluid Force: y surface x of fluid h(y) = depth | d | . F = pressure Area ( ( L(y) . y = (w depth (length width ( ( | . c | = (w h(y) (L(y) . y L(y) n . = w lim h(yi ) L(yi ) y F || || 0 i = 1 d = w h(y).L(y) dy F c where h(y) = depth of fluid at y and L(y) = horizontal length of the region at y. 5 Variable Depth 2.) Fluid Force on a Vertical Surface Example 2 p472 A vertical gate in a dam has the shape of an isosceles trapezoid 8 feet across the top and 6 feet across the bottom, with a height of 5 feet. What is the fluid force on the gate if the top of the gate is 4 feet below the surface of the water? 4 ft | 8 ft | | | 5 ft | | 6 ft | y x 4 ft | 4 ft 4 ft | h(y) = x 5 ft y | 3 ft 3 ft depth = h(y) = length = L(y) = y x 4 ft | 4 ft 4 ft ( , ) x 5 ft y | 3 ft 3 ft ( , ) 6 A vertical gate in a dam has the shape of an isosceles trapezoid 8 feet across the top and 6 feet across the bottom, with a height of 5 feet. What is the fluid force on the gate if the top of the gate is 4 feet below the surface of the water? 7 Variable Depth 3.) Fluid Force on a Vertical Surface A triangular plate, base 5 feet, height 6 feet, is submerged in water, vertex down, plane vertical, and 2 feet below the surface. Find the total force on one face of the plate. y water surface 2 ft {| 5 ft | | depth = h(y) = L(y) y 6 ft y | x Example 3 p473 4.) See CIRCULAR observation window 8 Assignment p474 #1, 3, 5-10 all, #11-19 odd 9.