Chapter 8 – Further Applications of
Integration
8.3 Applications to Physics and Engineering
1
8.3 Applications to Physics and Engineering
Erickson
Applications to Physics and Engineering

Among the many applications of integral calculus to
physics and engineering, we will consider two today:



Force due to water pressure
Center of mass
Our strategy is to break up the physical quantity into a
large number of small parts, approximate each small part,
add the results, take the limit, and then evaluate the
resulting integral.
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8.3 Applications to Physics and Engineering
Erickson
Hydrostatic Force and Pressure

Water pressure increases the father down your go because
the weight of the water above increases.

In general, we will submerge a thin horizontal plate with
area of A m2 in a fluid of density  kg/m3 at a depth d m
below the surface of the fluid. The fluid above the plate
has a volume V = Ad so its mass is m = V = Ad.

The force exerted by the fluid
on the plate is:
F = mg = gAd
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8.3 Applications to Physics and Engineering
Erickson
Hydrostatic Force and Pressure

The force exerted by the fluid on the plate is:
F = mg = gAd = Pd
Where A is the area of a thin plate
 is the fluid density in kg/m3
d is the depth in meters below the surface of the fluid.

The Pressure P on the plate is defined to be the force per unit area:
P = F/A = gd
The SI units for measuring pressure is newtons per square meter
which is called pascal. (1 N/m2 = 1 Pa)

Water’s weight density is
62.5 lb/ft2 or 1000kg/m3

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8.3 Applications to Physics and Engineering
Erickson
Force Exerted by a Fluid

The force F exerted by a fluid of a constant weightdensity w against a submerged vertical plane region from
y = c to y = d is
d
F  w h( y ) L( y )dy
c

Where w=  g, h(y) is the depth of the fluid and L(y) is
the horizontal length of the region at y.

Note: This integral comes from identifying the vertical
axis as the y-axis.
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8.3 Applications to Physics and Engineering
Erickson
Force Exerted by a Fluid

You can create a similar integral by choosing the vertical axis
to be x.

The force F exerted by a fluid of a constant weight-density w
against a submerged vertical plane region from x = a to x = b is
b
F  w h( x) L( x)dx
a
Where w=  g, h(x) is the depth of the fluid and L(x) is the
horizontal length of the region at x.

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8.3 Applications to Physics and Engineering
Erickson
Example 1 – pg. 560 #2

A tank is 8m long, 4m wide, 2m high, and contains
kerosene with density 820 kg/m3 to a depth of 1.5m. Find:
a)
b)
c)
7
The hydrostatic pressure on the bottom of the tank.
The hydrostatic force on the bottom.
The hydrostatic force on one end of the tank.
8.3 Applications to Physics and Engineering
Erickson
Example 2 – pg. 560

A vertical plate is submerged (or partially submerged) in
water and has the indicated shape. Explain how to
approximate the hydrostatic force against one side of the
plate by a Riemann sum. Then express the force as an
integral and evaluate it.
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8.3 Applications to Physics and Engineering
Erickson
Example 3 – pg. 560 #14

A vertical dam has a semicircular gate as shown in the
figure. Find the hydrostatic force against the gate.
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

We can find the point P on which a thin plate of any
given shape balances horizontally. This point is called the
center of mass or center of gravity of the plate.
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

The rod below will balance if m1d1=m2d2.

The numbers m1d1 and m2d2 are called the moments of the
masses.
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

If we put the rod along the x-axis, we will be able to solve
for point P, x
n
m1 x1  m2 x2
x

m1  m2

m x
i i
i 1
m
The numbers m1d1 and m2d2 are called the moments of the
masses
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

Momentn of the system about the origin
M   mi xi
i 1

Moment of
the system about the y-axis
n
M y   mi xi
i 1

Moment of
the system about the x-axis
n
M x   mi yi
i 1

In one dimensions, the coordinates of the center of mass are
given by
M M 
 
x, y  
, x
 m m 
13
y
8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass


Now we will consider a flat plate (lamina) with uniform
density  that occupies a region R of the plane. The
center of mass of the plate is called the centroid of R.
They symmetry principle says that if R is symmetric
about a line l, then



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The centroid of R lies on l.
Moments should be defined so that if the entire mass of a
region is concentrated at the center of mass, then its moments
remain unchanged.
The moment of the union of two non overlapping regions
should be the sum of the moments if the individual regions.
8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

So we have the moment of R about the y-axis:
 
n
M y  lim   xi f xi x
n 
i 1
b
   xf ( x)dx
a

The moment of R about the x-axis:
 
n
2
1
M x  lim  
f xi  x

n 
2
i 1


b
 f  x  

2
2
dx
a
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

The center of mass of the plate (the centroid of R) is
located at the point:
 x, y  where
b
1
x   xf ( x)dx
Aa
b
2
1
y
 f  x   dx

2A a
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8.3 Applications to Physics and Engineering
Erickson
Moments and Centers of Mass

If the region R is between two curves y = f (x) and
y=g(x), where f (x) ≥ g(x), as shown below, the then we
can say that the centroid of R is the point:
 x, y  where
b
1
x   x  f ( x )  g ( x )  dx
Aa
b


2
2
1
y
 f  x     g  x   dx

2A a
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8.3 Applications to Physics and Engineering
Erickson
Example 4 – pg. 561 #22

Point-masses mi are located on the x-axis as shown. Find
the moment M of the system about the origin and the
center of mass x .
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8.3 Applications to Physics and Engineering
Erickson
Example 5 – pg. 561 #24

The masses mi are located at the points Pi. Find the
moments Mx and My and the center of mass of the system.
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8.3 Applications to Physics and Engineering
Erickson
Example 6 – pg. 561 #32

Find the centroid of the region bounded by the given
curves.
y  x , x  y  2,
3
20
y 0
8.3 Applications to Physics and Engineering
Erickson
Theorem of Pappus

Let R be a plane region that lies entirely on one side of a
line l in the plane. If R is rotated about l, then the volume
of the resulting solid is the product of the area A of R and
the distance d traveled by the centroid of R.
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8.3 Applications to Physics and Engineering
Erickson
Example 7 – pg. 562

Use the Theorem of Pappus to find the volume of the
given solid.
45. A cone with the height h and base radius r
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8.3 Applications to Physics and Engineering
Erickson
Book Resources

Video Examples




More Videos





Example 1 – pg. 553
Example 3 – pg. 556
Example 7 – pg. 560
Problem on hydrostatic force and pressure – A
Problem on hydrostatic force and pressure – B
Moments and center of mass of a variable density planer lamina
Finding the center of mass
Wolfram Demonstrations


23
Center of Mass of n points
Theorem of Pappus on Surfaces of Revolution
8.3 Applications to Physics and Engineering
Erickson
Web Resources

http://youtu.be/H5RcfMIZ_yw

http://youtu.be/12MhraQo0TY

http://youtu.be/cXNmCaTod58

http://youtu.be/h8kMaW2q9EM

http://youtu.be/F2poHPZZBhE

http://youtu.be/NYUyHj3c1Xg

http://youtu.be/fJtxJv5sdqo
24

http://classic.hippocampus.org/cours
e_locator?course=General+Calculus
+II&lesson=57&topic=2&width=80
0&height=684&topicTitle=Work+d
one+on+a+fluid&skinPath=http%3
A%2F%2Fclassic.hippocampus.org
%2Fhippocampus.skins%2Fdefault

http://classic.hippocampus.org/cours
e_locator?course=General+Calculus
+II&lesson=62&topic=1&width=80
0&height=684&topicTitle=Center+o
f+mass+&+density&skinPath=http
%3A%2F%2Fclassic.hippocampus.
org%2Fhippocampus.skins%2Fdefa
ult
8.3 Applications to Physics and Engineering
Erickson