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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 405 "We can start to explore u
niform circular motion by making a few plots of\nthe path of a particl
e moving in a circle.  Let's begin by reminding ourselves\nof the para
metric description of a circle.  We define the x and y positions\nas f
unctions of time by noting that, in vector notation, the position vect
or\n_r_ has components r_x = r cos(theta) and r_y = r sin(theta).  The
refore,\nthe x and y positions are\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "x := r*cos(t*2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG*&%
\"rG\"\"\"-%$cosG6#,$*&%\"tGF'%#PiGF'\"\"#F'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "y := r*sin(t*2*Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG*&%\"rG\"\"\"-%$sinG6#,$*&%\"tGF'%#PiGF'\"\"#F'" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 578 "\nwhere we note that r is the \+
radius of the circular path and the expression 2*Pi*t allows\nus to sp
ecify the time as fractions of a period for the motion.  In other word
s,\nif t = 1 unit, that unit is the period of the motion and the x,y p
osition of the particle is\nthe same as it is at the beginning of the \+
motion (i.e. t=0 and t=1 give the same x,y positions).\nSince we are c
oncerned about the behavior as the particle moves around the\ncircular
 path, we can just pick a convenient radius of r = 1 to reduce typing.
  Let's\nthen make a parametric plot of the path for one period.\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r := 1:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "plot([x, y, t=0..1]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 89 "\nIt's more interesting to animate the path so we can see
 it as the particle goes around.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "animate
([cos(u*t*2*Pi), sin(u*t*2*Pi), t=0..1], u=0..1, color=red);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 1010 "\nTo find the association betwee
n arc-length and displacement of an item that travels\na circular cour
se, let's \"unroll\" the motion by having Maple show the distance trav
eled\nin a familiar format, namely along a straight-line course, while
 simultaneously showing\nthe circular path traveled.  We need to combi
ne some animations to do this.  First, we'll\nshow the unrolled path b
y animating the parametric equation of the circle.  We'll have\nour pa
rticle start at the bottom of the circle and we'll show the position o
f the particle\nrelative to the center of the circle by a line segment
 from the origin to the circle's perimeter.\nThen, we'll show the arc-
length traveled in a light blue (actually cyan) color as well as\nthe \+
straight-line length in the same color.  The path the particle will fo
llow is in yellow.\nTo make this all work we have to do the animations
 separately, then combine them all into\none plot at the end.  Again, \+
we use radius 1, but any value for the radius is fine.\n\nFirst the ci
rcular path,\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "P := animate([cos
(t), 2+sin(t), t=-Pi..Pi], u=0..6.283,color=magenta, view=[-1..7, 0..4
]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "\nNext, the line segment i
ndicating where the particle is\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
115 "Q := animate([-t*sin(-2*Pi*u/(2*Pi)), 2-t*cos(-2*Pi*u/(2*Pi)), t=
0..1], u=0..6.283, color=red, view=[-1..7, 0..4]):" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 79 "\nNow for the straight-line length indicating how
 far the particle has traveled\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 
"R := animate([u*t, 1, t=0..1], u=0..6.283, color=green, view=[-1..7, \+
0..4]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "\nFinally, put in the \+
arc-length the particle has traveled\n" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 134 "S := animate([cos(-Pi/2 + 2*t*u*Pi/(4*Pi^2)), 2+sin(-Pi/2+2*t
*u*Pi/(4*Pi^2)), t=0..2*Pi], u=0..6.283, color=cyan, view=[-1..7, 0..4
]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "\nNow we display them all
 together on the same plot.  You should go to the projections pull-dow
n menu\nand select constrained to see the circular path displayed as a
 circle.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display([P, Q, R, S])
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 642 "\nWe note that the path len
gth for one complete trip (in one period's time) is 6.283 or 2*Pi.  Th
us\nthe radian measure is usefully defined as the ratio between the ar
c-length and the radius.\nThat is to say, if we want the arc-length tr
aveled for a particular angular displacement,\nwe can state it as r * \+
delta(theta) where delta(theta) is the difference between the final\na
ngular position and the initial angular position.  Therefore, the spee
d of the particle as it goes\naround the circle is related to the angu
lar velocity as follows: d(theta)/dt = 1/r * v where\nv is the tangent
ial speed of the particle as it goes around the circular path." }}
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1 0 1 1 1803 }