chapter 8 – further applications of integration 8.3 applications to physics and engineering...
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Chapter 8 – Further Applications of Integration
8.3 Applications to Physics and Engineering
8.3 Applications to Physics and Engineering Erickson
8.3 Applications to Physics and Engineering2
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.
Erickson
8.3 Applications to Physics and Engineering3
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
Erickson
8.3 Applications to Physics and Engineering4
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
Erickson
8.3 Applications to Physics and Engineering5
Force Exerted by a Fluid The force F exerted by a fluid of a constant weight-density w
against a submerged vertical plane region from y = c to y = d is
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.
( ) ( )d
c
F w h y L y dy
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8.3 Applications to Physics and Engineering6
Force Exerted by a Fluid
Erickson
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
Where w= g, h(x) is the depth of the fluid and L(x) is the horizontal length of the region at x.
( ) ( )b
a
F w h x L x dx
8.3 Applications to Physics and Engineering7
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) The hydrostatic pressure on the bottom of the tank.
b) The hydrostatic force on the bottom.
c) The hydrostatic force on one end of the tank.
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8.3 Applications to Physics and Engineering8
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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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 Engineering10
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 Engineering11
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 Engineering12
Moments and Centers of Mass If we put the rod along the x-axis, we will be able to solve
for point P,
The numbers m1d1 and m2d2 are called the moments of the masses
x
11 1 2 2
1 2
n
i ii
m xm x m x
xm m m
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8.3 Applications to Physics and Engineering13
Moments and Centers of Mass Moment of the system about the origin
Moment of the system about the y-axis
Moment of the system about the x-axis
In one dimensions, the coordinates of the center of mass are given by
1
n
i ii
M m x
1
n
y i ii
M m x
1
n
x i ii
M m y
, ,y xM M
x ym m
Erickson
8.3 Applications to Physics and Engineering14
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 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.
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8.3 Applications to Physics and Engineering15
Moments and Centers of Mass So we have the moment of R about the y-axis:
The moment of R about the x-axis:
1
lim
( )
n
y i ini
b
a
M x f x x
xf x dx
2
1
2
1lim
2
2
n
x in
i
b
a
M f x x
f x dx
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8.3 Applications to Physics and Engineering16
Moments and Centers of Mass The center of mass of the plate (the centroid of R) is
located at the point:
2
, where
1( )
1
2
b
a
b
a
x y
x xf x dxA
y f x dxA
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8.3 Applications to Physics and Engineering17
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:
2 2
, where
1( ) ( )
1
2
b
a
b
a
x y
x x f x g x dxA
y f x g x dxA
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8.3 Applications to Physics and Engineering18
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 Engineering19
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 Engineering20
Example 6 – pg. 561 #32 Find the centroid of the region bounded by the given
curves.3 , 2, 0y x x y y
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8.3 Applications to Physics and Engineering21
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 Engineering22
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 Engineering23
Book Resources
Erickson
Video Examples Example 1 – pg. 553 Example 3 – pg. 556 Example 7 – pg. 560
More Videos 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 Center of Mass of n points Theorem of Pappus on Surfaces of Revolution
8.3 Applications to Physics and Engineering24
Web Resources
Erickson
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
http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=57&topic=2&width=800&height=684&topicTitle=Work+done+on+a+fluid&skinPath=http%3A%2F%2Fclassic.hippocampus.org%2Fhippocampus.skins%2Fdefault
http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=62&topic=1&width=800&height=684&topicTitle=Center+of+mass+&+density&skinPath=http%3A%2F%2Fclassic.hippocampus.org%2Fhippocampus.skins%2Fdefault