8.3 Applications to Physics and Engineering In this section, we will discuss only one application of...
6
8.3 Applications to Physics and Engineering In this section, we will discuss only one application of integral calculus to physics and engineering and this topic is: The Center of Mass of Planar Lamina Consider a thin flat plate of material with uniform density called a planar laminar. We think of center of mass as its balancing point. y x C ,
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B. R lies between on the interval [a, b] where x ab y Mass of R Centroid of R:
Transcript of 8.3 Applications to Physics and Engineering In this section, we will discuss only one application of...
8.3 Applications to Physicsand Engineering
In this section, we will discuss only one application of integral calculus to physics and engineering and this topic is:
The Center of Mass of Planar Lamina
Consider a thin flat plate of material with uniform density called a planar laminar. We think of center of mass as its balancing point.
yxC ,
Centroid of a Plane RegionConsider a flat plate with uniform density that occupies a region R of the plane
A.
yxC ,
)(xfy
a
R
bx
y Moment of R about the y-axis:
b
ay dxxxfM )(
Moment of R about the x-axis:
b
ax dxxfM 2)(21
Mass of Plate = (Density)(Area)
b
ax
b
a
y
dxxfAm
My
dxxxfAm
Mx
2)(211
)(1
b
adxxfAm )(
Centroid of R: yxC ,
Note: If a lamina has the shape of a region that has an axis of symmetry, then the center of mass must lie on that axis.
B. R lies between on the interval [a, b] where )()( xgxf )( & )( xgyxfy
x
yxC ,
)(xfy
)(xgy
a b
y
Mass of R dxxgxfAmb
a )()(
Centroid of R:
dxxgxfA
y
dxxgxfxA
x
yxC
b
a
b
a
22 )()(211
)()(1 ,
Examples:1) Find the centroid of the region bounded by the curves.
exyxy ,0 ),ln(
1 e
x
y
yxC ,
)ln(xy
Solutions:
e
dxxA1
)ln(xvdx
xdu
dxdvxu
1
)ln(let
110)ln()ln( 111 eexxxdxxxA eee
41
41
41
21
4)ln(
21
21)ln(
21
)ln(11
)(1
222
1
22
11
2
1
eee
xxx
xdxxx
dxxx
dxxxfA
x
e
ee
e
b
a
2
x1
)ln(2x
xx D I
22222
21
2)ln(2)(ln21
)(ln21
11
)(211
12
1
2
2
eeee
xxxx
dxx
dxxfA
y
e
e
b
a
xx
x
xxxx
21 )ln(2
)ln(2
1 )(ln 2
D I
22,
41 ,
2 eeCyxC
2) Find the center of mass of a semicircular plate of radius r.22 xry
yxC ,
(-r, 0) (r, 0)
By principle of symmetry, center of mass must lie on the y-axis.
0
,0
x
yCx
y
34
341
331
311
21
2
12
re whe
)(211
3
2
33
33
2
32
222
2
222
2
2
2
rrr
rr
rr
r
xxrr
dxxrr
dxxrr
y
rA
dxxfA
y
r
r
r
r
r
r
b
a
34 ,0 , rCyxC
3) Find the centroid bounded by the given curves.2 ,4 2 xyxy
Points of intersectionSolutions: 2or 10224 22 xxxxxx
24 xy 2xy
(-2,0)
(1, 3) yxC ,
x
y
29
3824
31
212
322
224
1
2
32
1
2
21
2
2
xxx
dxxxdxxxA
21
4392
292
2492
)()(1
1
2
432
1
2
2
1
2
2
xxx
dxxxx
dxxxx
dxxgxfxA
xb
a
5121223
591
124991
2421
92
)()(211
1
2
235
1
2
24
1
2
222
22
xxxx
dxxxx
dxxx
dxxgxfA
yb
a
512 ,
21 , CyxC
