Fluid Force
Fluid Pressure (Definition)
We define the fluid pressure
P to be equal to the weight density
w times
the depth of the fluid h.
|
P = wh |
Pascal's Principle
The pressure is equal in all directions
Since
force
Pressure =
area
we have
|
F = PA |
Force on a Submerged Rectangular Sheet
Example
Suppose that a glass sheet lies parallel to and five feet below the surface of a lake
has dimensions 2 by 3. Then the total pressure on the sheet is
P = wh = (62.4)(5) = 312
The total force on the plate is
PA = (312)(6) = 1872
Force on a Vertical Surface
Example
A 3 x 2 square window on the new Disney Cruise liner
is to be built so that top of the window is four feet below the surface of
the water. What total force will the window be subjected to?
Solution
We take horizontal cross sections. Letting y
be the distance from the top of the window to the cross section, we have
DF = 62.4(4 + y)(D A) = 62.4(4 + y)(3)(Dy)
Hence the total force is
= 1123.2 pounds
In general:
|
Example
A radius 2 feet circular portal is vertically submerged so that its center is 20 feet below the surface of the water. Find the total force of the water on the portal.
Solution
We write s in terms of z by the Pythagorean theorem:
This horizontal cross-section has area
DA =
2sDz
The
depth at this cross-section is
h = 20 + z
We put this all together to find the force
We
recognize the first integral as the area of the semi-circle of radius 2.
For the second integral, let
u = 4 - z2
du = -2zdz
Notice that when z = -2, u = 0 and when z = 2, u = 0 also. Any integral
with lower and upper limits the same is always zero. Hence the force
is
F = 40p22
+ 0 = 160p
Exercise:
In the VIP suite, the glass window is in the shape of an equilateral triangle with diagonal length 2 such that the top of the triangle is submerged 5 feet below the water. What total force will this window be subjected to?
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