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{SECT 0 {PARA 261 "" 0 "" {TEXT -1 9 "Centroids" }}{PARA 257 "" 0 ""
{TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1 21 "School of Math
ematics" }}{PARA 259 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 260 "
" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 336 " This worksheet is concerned \+
with centroids for curves and regions in two dimensions. To help intui
tion, questions are asked about how centroids move as the curves or re
gions grow or shrink. The important idea is that the coordinates of th
e centroid are given by quotients of integrals. For example, for a sur
face in two dimensions," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 35 " xbar = " }{XPPEDIT 18 0
"`x dA`/`dA`" "6#*&%%x~dAG\"\"\"%#dAG!\"\"" }{TEXT -1 16 " and yba
r = " }{XPPEDIT 18 0 "`y dA`/`dA`" "6#*&%%y~dAG\"\"\"%#dAG!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 56 "The centroid of a half \+
washer morphing to a half circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 91 " We imagine a half-washer with inner \+
radius a and outer radius b. We find the centroid " }{XPPEDIT 18 0 "[x
[a],y[a]]" "6#7$&%\"xG6#%\"aG&%\"yG6#F'" }{TEXT -1 110 " of this regio
n of the plane and take the limit as a -> b. The result is compared wi
th the centroid of a half-" }{TEXT 259 6 "circle" }{TEXT -1 13 " of ra
dius b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 18 "Drawing the figure" }}{PARA 0 "" 0 "" {TEXT -1 75 " T
he planer region for this problem is a half washer with inner radius \+
" }{TEXT 256 1 "a" }{TEXT -1 18 " and outer radius " }{TEXT 257 1 "b"
}{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "plot(\{[r,0,r=2/3..1
],[r,Pi,r=2/3..1],\n [1,theta,theta=0..Pi],[2/3,theta,theta=0..Pi
]\},\ncoords=polar,scaling=constrained,axes=NONE,color=BLACK);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Computing xbar and ybar
" }}{PARA 0 "" 0 "" {TEXT -1 96 " The coordinates for the center o
f mass are computed two ways. The first method is analytic." }}{SECT
1 {PARA 4 "" 0 "" {TEXT -1 16 "Analytic method." }}{PARA 0 "" 0 ""
{TEXT -1 118 "In the analytic method for computing the center of mass \+
for the half washer, we use the integral formulas given above." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "xbar:=int(int(r*cos(t)*r,r=
a..b),t=0..Pi)/int(int(r,r=a..b),t=0..Pi);\nybar:=int(int(r*sin(t)*r,r
=a..b),t=0..Pi)/int(int(r,r=a..b),t=0..Pi);\nybar:=simplify(ybar);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Geometric method" }}
{PARA 0 "" 0 "" {TEXT -1 80 " In this geometric method, we think o
f two regions: a half-disk with radius " }{TEXT 258 1 "a" }{TEXT -1
357 " and the half washer with inner radius a and outer radius b. Thin
k of the union of these two sets. The centers of mass of these two are
related by an algebraic formula to the center of mass of their union:
a disk with radius b. If we compute the simplier center of mass of a
half disk, this equation can be solved for the center of mass of the \+
half washer." }}{PARA 0 "" 0 "" {TEXT -1 65 " The center of mass o
f a half disk with radius b is computed." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 141 "xdisk:=int(int(r*cos(t)*r,r=0..b),t=0..Pi)/int(int(r
,r=0..b),t=0..Pi);\nydisk:=int(int(r*sin(t)*r,r=0..b),t=0..Pi)/int(int
(r,r=0..b),t=0..Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 124 "Thus, the y coo
rdinate for the center of mass of the washer can be computed as the so
lution, yy, for the following equation." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 67 "eq:=4/3*a/Pi*(Pi*a^2/2)+yy*(Pi*b^2/2-Pi*a^2/2)=4/3*b/
Pi*(Pi*b^2/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(eq,
yy);" }}}}{PARA 0 "" 0 "" {TEXT -1 89 "Check that the answer by geomet
ric methods is the same as the answer by analytic methods." }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Taking \+
the limit." }}{PARA 0 "" 0 "" {TEXT -1 53 " We compute the limit o
f xbar and ybar as a -> b." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
34 "[limit(xbar,a=b),limit(ybar,a=b)];" }}}{PARA 0 "" 0 "" {TEXT -1 1
" " }}{PARA 0 "" 0 "" {TEXT -1 105 " As a -> b, the half washer ha
s limit a half circle. We compute the center of mass for a half circle
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "int(b*cos(t)*b,t=0..Pi)
/(Pi*b);\nint(b*sin(t)*b,t=0..Pi)/(Pi*b);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 16 "The question is:"
}}{PARA 0 "" 0 "" {TEXT -1 110 " Is the center of mass of
the half circle equal to the limit of the center of mass of the washe
r?" }}{PARA 0 "" 0 "" {TEXT -1 57 "A quick look at the results shows t
hat the answer is yes." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 3 "" 0 "" {TEXT -1 25 "Exercises for the student" }}{PARA 0 ""
0 "" {TEXT -1 125 "(1) Show that as a -> 0, the center of mass for the
half-washer has limit the center of mass for the half-disk with radiu
s b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 362 "
(2) Consider a rectangle with height b and length 2b. Cut-out a notch
with a similar shape, but with height a and length 2a, a < b. Find th
e center of mass of this region. Take the limit as a -> b. Does this l
imit give the center of mass of a wire which is the top and two sides \+
of a rectangle? (Hint: Geometric methods will work for all parts of th
is problem.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot([[1,0]
,[1,1],[-1,1],[-1,0],[-2/3,0],[-2/3,2/3],[2/3,2/3],\n [2/3,0],[1,0
]],scaling=constrained,axes=none);" }}}}{EXCHG {PARA 0 "> " 0 ""
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