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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Gabriel, Gabe, and Chris" 
}}{PARA 0 "" 0 "" {TEXT -1 9 "#5. p.203" }}{PARA 0 "" 0 "" {TEXT -1 
421 "We are given a mass of 5kg is attached to a spring hanging from t
he ceiling, which stretches it .5m when it comes to rest at equilibriu
m.  The mass is then pulled down .1m below the equilibrium point and g
iven upward velocity of .1m/sec.  We are then asked to determine the e
quation for the simple harmonic motion of the mass and also the time t
hat the mass will first reach it minimum height after being set into m
otion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 233 "eq1 is the general differential e
quation for simple harmonic motion undamped and no external forces. Th
e model differential equation for a weight attached to a spring is mx
\"+bx'+kx=f(t). In this case, m=5kg, b=0, k=mg/l, and f(t)=0. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eq1:=diff(x(t),t$2) + w^2 * \+
x(t) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&-%%diffG6$-%\"x
G6#%\"tG-%\"$G6$F-\"\"#\"\"\"*&)%\"wGF1\"\"\"F*F2F2\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 83 "Here we solved the general differential e
quation (eq1) and defined it as solution. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "solution:=dsolve(eq1,x(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)solutionG/-%\"xG6#%\"tG,&*&%$_C1G\"\"\"-%$cosG6#*&%
\"wGF-F)F-F-F-*&%$_C2GF--%$sinGF0F-F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "Here we defined _C1 and _C2." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "_C1:=A*sin(phi); _C2:=A*cos(phi);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$_C1G*&%\"AG\"\"\"-%$sinG6#%$phiGF'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$_C2G*&%\"AG\"\"\"-%$cosG6#%$phiGF'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This is our simple harmonic equati
on with _C1 and _C2 plugged in." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "solution;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*(
%\"AG\"\"\"-%$sinG6#%$phiGF+-%$cosG6#*&%\"wGF+F'F+F+F+*(F*\"\"\"-F1F.F
+-F-F2F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Here we used trig i
dentities to simplify the solution." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "eq2:=combine(solution,trig);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/-%\"xG6#%\"tG*&%\"AG\"\"\"-%$sinG6#,&%$phiGF,*&
%\"wGF,F)F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Here we defin
ed k. k is mg/l. l is the length that the mass stretches the spring. k
 is our spring constant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"k:=5*9.8/.5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+++++)*!\")
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Here we defined w, which is o
ur angular frequency. w=sqrt(k/m)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "w:=sqrt(k/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
wG$\"+C()=FW!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "This is our \+
simple harmonic motion equation with the given conditions plugged into
 it, which is the first thing the problem asked for." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 74 "eq1:=dsolve(\{diff(x(t),t$2) + w^2 * x(t)
 = 0, x(0)=.1, D(x)(0)=.1\}, x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$eq1G/-%\"xG6#%\"tG,&-%$sinG6#,$F)$\"+C()=FW!\"*$\"+d(p(eA!#6-%$cos
GF-$\"+++++5!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Here we unappl
ied the equation, so it could be plotted." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "y1:=unapply(rhs(eq1),t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y1GR6#%\"tG6\"6$%)operatorG%&arrowGF(,&-%$sinG6#,$9$
$\"+C()=FW!\"*$\"+d(p(eA!#6-%$cosGF/$\"+++++5!#5F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "plot(y1(t),t=0..4*Pi/w);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "A is our ampl
itude for our equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "
A:=evalf(sqrt((.02258769757)^2+.1^2));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG$\"+kG>D5!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "phi i
s the phase angle for our equation." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "phi:=evalf(arctan(.1/.02258769757));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$phiG$\"+Btk[8!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "Here we took the
 derivative of our simple harmonic motion equation and set it equal to
 zero. This allowed us to find maximums and minimums of our equation.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "deriv:=diff(A*sin(phi+w
*t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG/,$-%$cosG6#,&$
\"+Btk[8!\"*\"\"\"%\"tG$\"+C()=FWF-$\"+(GA(QX!#5\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "This is the time at which our first minim
um occured." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(deriv
,t=.5..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fz\"zf(!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 24 }
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