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6.7 Fluid and Fluid

The pressure at any point on a dam depends only on how Fact #1: far below the 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 is

Pressure = p = w .h where w is the ‐density of the liquid per unit of .

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.

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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

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