Transcript Calc07_5day2

7.5 Fluid Pressure and Forces
Greg Kelly, Hanford High School, Richland, Washington
What is the force on the bottom of
the aquarium?
2 ft
1 ft
3 ft
Force  weight of water
 density  volume
lb
 62.5 3  2 ft  3 ft 1 ft
ft
 375 lb

If we had a 1 ft x 3 ft plate on
the bottom of a 2 ft deep
wading pool, the force on the
plate is equal to the weight of
the water above the plate.
lb
62.5 3 2 ft 3 ft 1 ft  375 lb
ft
density depth
pressure
area
All the other water in the
pool doesn’t affect the
answer!

What is the force on the front face
of the aquarium?
2 ft
Depth (and pressure) are not constant.
If we consider a very thin horizontal
strip, the depth doesn’t change much,
and neither does the pressure.
3 ft
1 ft
Fy  62.5  y  3 dy
depth
density
area
0
y
2
2 ft
dy
2
3 ft
F   62.5  y  3 dy
0
It is just a
coincidence
that this
matches the
first answer!
2
187.5 2
F
y  375 lb
2
0

A flat plate is submerged vertically
We could have put
as shown. (It is a window in the
the origin at the
surface, but the math shark pool at the city aquarium.)
was easier this way.
2 ft
Find the force on one side of the
plate.
yx
x y
3 ft
6 ft
Depth of strip:  5  y 
Length of strip: 2 x  2 y
Area of strip: 2 y dy
3
Fy  62.5  5  y  2 y dy
F  125 5 y  y 2 dy
density depth area
5 2 1 3
F  125  y  y 
3 0
2
0
3
F   62.5  5  y  2 y dy
3
0
F  1687.5 lb

Normal Distribution:
13.5%
34%
3 2  
2.35%
 2 3
68%
95%
99.7%
“68, 95, 99.7 rule”
For many real-life events, a
frequency distribution plot
appears in the shape of a
“normal curve”.
Examples:
The mean  (or x ) is in the
middle
of theofcurve.
heights
18 yr. The
old men
shape of the curve is
standardized
scores
determined
by thetest
standard
deviation  .
lengths of pregnancies
 mu
time
for corn to pop
x

x-bar
sigma

Normal Distribution:
13.5%
34%
3 2  
2.35%
The area under the curve
from a to b represents the
probability of an event
occurring within that range.
 2 3
“68, 95, 99.7 rule”
In Algebra 2 we used z-scores and a table of values to
determine probabilities. If we know the equation of the
curve we can use calculus (and our calculator) to determine
probabilities:
Normal Probability
Density Function:
(Gaussian curve)
2
2

x


/
2



1


f  x 
e
 2

Normal Distribution:
The good news is that you do not have to memorize this
equation!
Example 6 on page 406 shows how you could integrate
this function to predict probabilities.
In real life, statisticians rarely see this function. They use
computer programs or graphing calculators with statistics
software to draw the curve or predict the probabilities.
Normal Probability
Density Function:
(Gaussian curve)
2
2

x


/
2



1


f  x 
e
 2
