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