Nick Berger, Tim Roorda, Juan Burgos and Andrew Eisterhold
3D Center of Massrestart;with(Maplets[Tools]): with(Maplets[Elements]): with(plots):
with(RandomTools):# with(plottools):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):randx1:=rand(-1..3):
randx2:=rand(1..4):
randtype:=rand(0..1):ask:=proc()
global f, g, h, a, b, M, Mval, Myz, Mxz, Mxy, Myzval, Mxzval, Mxyval, rho, X, Y, Z, intType, u, v, z, rho1;
local p1, fmin, gmax, hmin, rhomin, dv, pad, f1, g1, h1, f2, g2, p2, ptop, pbot, pfrt, pbck, pright, pleft;
rho:=Generate(polynom(integer(range=-3..3), {u,v,z}, degree=1));
intType:=randtype();
f:=randpoly(u, coeffs=rand(-3..3), degree=2, terms=2);
g:=randpoly(u, coeffs=rand(-3..3), degree=2, terms=2);
h:=randpoly([u,v], coeffs=rand(-3..3), degree=2, terms=2);
while h=0 do
h:=randpoly([u,v], coeffs=rand(1..3), degree=2, terms=2);
end do;
a:=randx1();
b:=a+randx2();
fmin:=minimize(f, u=a..b);
gmax:=maximize(g, u=a..b);
dv:=fmin-gmax;
if dv<0 then
f:=f-dv;
end if;
hmin:=minimize(h, u=a..b, v=minimize(g,u=a..b)..maximize(f,u=a..b));
if hmin<0 then
h:=h-hmin;
end if;
rhomin:=minimize(rho, u=a..b, v=minimize(g,u=a..b)..maximize(f,u=a..b),
z=0..maximize(h,u=a..b, v=minimize(g,u=a..b)..maximize(f,u=a..b)));
if rhomin<0 then
rho:=rho-rhomin;
end if;
pad:=(b-a)/5;
if intType=0 then
f1:=subs(u=x,f); g1:=subs(u=x,g);
h1:=subs({u=x,v=y},h);
rho1:=subs({u=x,v=y},rho);
M:= Int(Int(Int( rho1,z=0..h1),y=g1..f1),x=a..b);
Myz:=Int(Int(Int(x*rho1,z=0..h1),y=g1..f1),x=a..b);
Mxz:=Int(Int(Int(y*rho1,z=0..h1),y=g1..f1),x=a..b);
Mxy:=Int(Int(Int(z*rho1,z=0..h1),y=g1..f1),x=a..b);
Set(question=sprintf("Find the center of mass of the solid\134nbelow z = %a above the region\134nbetween y = %a and y = %a\134nfor the interval %a <= x <= %a\134nwith density %a.",
h1, f1, g1, a, b, rho1));
ptop:= plot3d([x,y,h1], x=a..b, y=g1..f1, color=green):
pbot:= plot3d([x,y,0], x=a..b, y=g1..f1, color=green):
pbck:= plot3d([a,y,z], y=eval(g1,x=a)..eval(f1,x=a), z=0..eval(h1,x=a), color=magenta):
pfrt:= plot3d([b,y,z], y=eval(g1,x=b)..eval(f1,x=b), z=0..eval(h1,x=b), color=magenta):
pright:=plot3d([x,f1,z], x=a..b, z=0..eval(h1,y=f1), color=blue):
pleft:= plot3d([x,g1,z], x=a..b, z=0..eval(h1,y=g1), color=blue):
Set(graph3d=display([ptop,pbot,pbck,pfrt,pleft,pright], orientation=[-90,0], axes=normal));
else
f1:=subs(u=y,f); g1:=subs(u=y,g);
h1:=subs({u=y,v=x},h);
rho1:=subs({u=y,v=x},rho);
M:= Int(Int(Int( rho1,z=0..h1),x=g1..f1),y=a..b);
Myz:=Int(Int(Int(x*rho1,z=0..h1),x=g1..f1),y=a..b);
Mxz:=Int(Int(Int(y*rho1,z=0..h1),x=g1..f1),y=a..b);
Mxy:=Int(Int(Int(z*rho1,z=0..h1),x=g1..f1),y=a..b);
Set(question=sprintf("Find the center of mass of the solid\134nbelow z = %a above the region\134nbetween x = %a and x = %a\134nfor the interval %a <= y <= %a\134nwith density %a.",
h1, f1, g1, a, b, rho1));
ptop:= plot3d([x,y,h1], y=a..b, x=g1..f1, color=green):
pbot:= plot3d([x,y,0], y=a..b, x=g1..f1, color=green):
pbck:= plot3d([x,a,z], x=eval(g1,y=a)..eval(f1,y=a), z=0..eval(h1,y=a), color=magenta):
pfrt:= plot3d([x,b,z], x=eval(g1,y=b)..eval(f1,y=b), z=0..eval(h1,y=b), color=magenta):
pright:=plot3d([f1,y,z], y=a..b, z=0..eval(h1,x=f1), color=blue):
pleft:= plot3d([g1,y,z], y=a..b, z=0..eval(h1,x=g1), color=blue):
Set(graph3d=display([ptop,pbot,pbck,pfrt,pleft,pright], orientation=[-90,0], axes=normal));
end if;
Mval:=value(M);
Myzval:=value(Myz);
Mxzval:=value(Mxz);
Mxyval:=value(Mxy);
X:=value(Myz/M);
Y:=value(Mxz/M);
Z:=value(Mxy/M);
Set(MInt="");
Set(MVal="");
Set(MxyVal="");
Set(MyzVal="");
Set(MxzVal="");
Set(MxyInt="");
Set(MyzInt="");
Set(MxzInt="");
Set(XCoord="");
Set(YCoord="");
Set(ZCoord="");
end proc:checkM:=proc()
global Mval;
local userM, userMval;
userM:=Get(MInt::function):
userMval:=Get(MVal::realcons):
if userM=M and userMval=Mval then
Set(reply="The mass integral is correct!")
elif userMConv=M then
Set(reply="The mass integral is correct, but the value is wrong. Try again.")
else
Set(reply="The mass integral is wrong. Try again.")
end if;
end proc:hintM:=proc()
if intType=0 then
Set(MInt="Int(Int(Int(rho,z = 0 .. h(x,y)),y = f(x) .. g(x)),x = a .. b)")
else
Set(MInt="Int(Int(Int(rho,z = 0 .. h(x,y)),x = f(y) .. g(y)),y = a .. b)")
end if;
end proc:showM:=proc()
global M, Mval;
Set(MInt=M):
Set(MVal=Mval):
end proc:checkMyz := proc ()
local userMyz, userMyzval;
global Myzval;
userMyz := Get(MyzInt::realcons);
userMyzval := Get(MyzVal::realcons);
if userMyz=Myz and userMyzval=Myzval then
Set(reply="The Myz integral and value are correct!")
elif userMyzConv=Myz then
Set(reply="The Myz integral is correct, but the value is wrong. Try again.")
else
Set(reply="The Myz integral is wrong. Try again.")
end if;
end proc:hintMyz := proc ()
if intType=0 then
Set(MyzInt = "Int(Int(Int(x*rho,z = 0 .. h(x,y)),y = f(x) .. g(x)),x = a .. b)")
else
Set(MyzInt = "Int(Int(Int(x*rho,z = 0 .. h(x,y)),x = f(y) .. g(y)),y = a .. b)")
end if;
end proc:showMyz:=proc()
global Myz, Myzval;
Set(MyzInt = Myz);
Set(MyzVal = Myzval)
end proc:checkX:=proc()
local userX;
userX:=Get(XCoord::realcons);
if userX=X then
Set(reply="Correct x center of mass!")
else
Set(reply="Incorrect x center of mass. Try again.")
end if;
end proc:hintX:=proc ()
Set(XCoord="Myz/M")
end proc:showX:=proc ()
Set(XCoord=X)
end proc:checkMxz:=proc()
local userMxz, userMxzval:
global Mxzval:
userMxz:=Get(MxzInt::realcons):
userMxzval:=Get(MxzVal::realcons):
if userMxz=Mxz and userMxzval=Mxzval then
Set(reply="The Mxz integral and value are correct!"):
elif userMxzConv=Mxz then
Set(reply="The Mxz integral is correct, but the value is wrong. Try again."):
else
Set(reply="The Mxz integral is wrong. Try again."):
end if:
end proc:hintMxz := proc ()
if intType=0 then
Set(MxzInt = "Int(Int(Int(y*rho,z = 0 .. h(x,y)),y = f(x) .. g(x)),x = a .. b)")
else
Set(MxzInt = "Int(Int(Int(y*rho,z = 0 .. h(x,y)),x = f(y) .. g(y)),y = a .. b)")
end if;
end proc:showMxz := proc ()
global Mxz, Mxzval;
Set(MxzInt = Mxz);
Set(MxzVal = Mxzval)
end proc:checkY:=proc ()
local userY;
userY:=Get(YCoord::realcons);
if userY=Y then
Set(reply="Correct y center of mass!")
else
Set(reply="Incorrect y center of mass. Try again.")
end if;
end proc:hintY := proc ()
Set(YCoord = "Mxz/M")
end proc:showY := proc ()
Set(YCoord = Y)
end proc:checkMxy := proc ()
local userMxy, userMxyval;
global Mxyval;
userMxy := Get(MxyInt::realcons);
userMxyval := Get(MxyVal::realcons);
if userMxy=Mxy and userMxyval=Mxyval then
Set(reply="The Mxy integral and value are correct!")
elif userMxyConv=Mxy then
Set(reply="The Mxy integral is correct, but the value is wrong. Try again.")
else
Set(reply="The Mxy integral is wrong. Try again.")
end if;
end proc:hintMxy := proc ()
if intType=0 then
Set(MxyInt = "Int(Int(Int(z*rho,z = 0 .. h(x,y)),y = f(x) .. g(x)),x = a .. b)")
else
Set(MxyInt = "Int(Int(Int(z*rho,z = 0 .. h(x,y)),x = f(y) .. g(y)),y = a .. b)")
end if;
end proc:showMxy:=proc()
global Mxy, Mxyval;
Set(MxyInt = Mxy);
Set(MxyVal = Mxyval)
end proc:checkZ:=proc ()
local userZ;
userZ:=Get(ZCoord::realcons);
if userZ=Z then
Set(reply="Correct z center of mass!")
else
Set(reply="Incorrect z center of mass. Try again.")
end if;
end proc:hintZ := proc ()
Set(ZCoord = "Mxy/M")
end proc:showZ := proc ()
Set(ZCoord = Z)
end proc:Preview:=proc(TFname)
global CurrentTFname;
local expr;
options `Copyright 2004-07, Philip B. Yasskin and Douglas Meade, Maplets for Calculus`;
if Get(TFname)="" then
Set(reply="Please enter an expression to preview.");
return;
end if;
CurrentTFname:=TFname;
expr:=Get(TFname::anything, corrections=true, update=true);
Set(TFPreview=expr);
Set(MMLPreview=MathML[Export](expr));
Set(ARunPreview(run)=true);
end proc:Update_Preview:=proc()
global CurrentTFname;
local expr;
options `Copyright 2004-07, Philip B. Yasskin and Douglas Meade, Maplets for Calculus`;
if Get(TFPreview)="" then
Set(reply="Please enter an expression to preview.");
return;
end if;
expr:=Get(TFPreview::anything, corrections=true, update=true);
Set(MMLPreview=MathML[Export](expr));
Set(CurrentTFname=expr);
end proc:CenterOfMass:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN] (title="Center of Mass of a Plate with Variable Density",
[ valign=none, inset=0, spacing=0,
[ Button("New Problem", Evaluate(function = "ask"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ halign=none, valign=none,
[ halign=none, valign=none, inset=0,
[ halign=none,
HorizontalGlue(),
TextBox[question](width=45, height=5, editable=false),
HorizontalGlue()
],
BoxColumn( valign=none, inset=0, spacing=0, border=true,
caption="Find the total mass, M. Enter the integral and value.",
[ halign=none,
"M = ",
TextBox[MInt](width = 45, height=2, background=turquoise)
],
[ halign=none,
" = ",
TextField[MVal](width=10, background=turquoise),
HorizontalGlue(),
Button("Template", Evaluate(function="hintM"), background=lightorange),
Button("Preview", onclick=APreM, background=yellow),
Button("Check", Evaluate(function="checkM"), background=green),
Button("Show", Evaluate(function="showM"), background=lightorange),
HorizontalGlue()
]
)
],
[ border=true,
caption="Rotate the plot with your mouse to see the solid.",
Plotter[graph3d](plot3d(0,x=0..6,y=0..6,axes=normal), width=250, height=200)
]
],
[ [ valign=none, inset=0, spacing=0, border=true,
caption="Find the moment about the yz-plane (integral and value):",
[ halign=none,
"Myz = ",
TextBox[MyzInt](width=45, height=2, background=turquoise)
],
[ halign=none,
" =",
TextField[MyzVal](width=10, background=turquoise),
HorizontalGlue(),
Button("Template", Evaluate(function="hintMyz"), background=lightorange),
Button("Preview", onclick=APreMyz, background=yellow),
Button("Check", Evaluate(function="checkMyz"), background=green),
Button("Show", Evaluate(function="showMyz"), background=lightorange),
HorizontalGlue()
]
],
[ valign=none, inset=0, spacing=0, border=true,
caption="Find the x-coordinate of the center of mass:",
[ halign=none,
HorizontalGlue(),
"X = ",
TextField[XCoord](width=20, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintX"), background=lightorange),
Button("Check", Evaluate(function="checkX"), background=green),
Button("Show", Evaluate(function="showX"), background=lightorange),
HorizontalGlue()
]
]
],
[
[ valign=none, inset=0, spacing=0, border=true,
caption="Find the moment about the xz-plane (integral and value):",
[ halign=none,
"Mxz = ",
TextBox[MxzInt](width=45, height=2, background=turquoise)
],
[ halign=none,
" =",
TextField[MxzVal](width=10, background=turquoise),
HorizontalGlue(),
Button("Template", Evaluate(function="hintMxz"), background=lightorange),
Button("Preview", onclick=APreMxz, background=yellow),
Button("Check", Evaluate(function="checkMxz"), background=green),
Button("Show", Evaluate(function="showMxz"), background=lightorange),
HorizontalGlue()
]
],
[ valign=none, inset=0, spacing=0, border=true,
caption="Find the y-coordinate of the center of mass:",
[ halign=none,
HorizontalGlue(),
"Y = ",
TextField[YCoord](width=20, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintY"), background=lightorange),
Button("Check", Evaluate(function="checkY"), background=green),
Button("Show", Evaluate(function="showY"), background=lightorange),
HorizontalGlue()
]
]
],
[
[ valign=none, inset=0, spacing=0, border=true,
caption="Find the moment about the xy-plane (integral and value):",
[ halign=none,
"Mxy = ",
TextBox[MxyInt](width=45, height=2, background=turquoise)
],
[ halign=none,
" =",
TextField[MxyVal](width=10, background=turquoise),
HorizontalGlue(),
Button("Template", Evaluate(function="hintMxy"), background=lightorange),
Button("Preview", onclick=APreMxy, background=yellow),
Button("Check", Evaluate(function="checkMxy"), background=green),
Button("Show", Evaluate(function="showMxy"), background=lightorange),
HorizontalGlue()
]
],
[ valign=none, inset=0, spacing=0, border=true,
caption="Find the z-coordinate of the center of mass:",
[ halign=none,
HorizontalGlue(),
"Z = ",
TextField[ZCoord](width=20, background=turquoise),
HorizontalGlue()
],
[ halign=none,
HorizontalGlue(),
Button("Template", Evaluate(function="hintZ"), background=lightorange),
Button("Check", Evaluate(function="checkZ"), background=green),
Button("Show", Evaluate(function="showZ"), background=lightorange),
HorizontalGlue()
]
]
],
[ TextField[reply](width=60)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Sarah Mannen, Nick Berger, Tim Roorda, Juan Burgos and Andrew Eisterhold", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]),
Window[PREVIEW]( title="Preview", defaultbutton=UpdatePreview,
[ halign=none,
[ TextField[TFPreview](width=50, background=turquoise),
Button[UpdatePreview]("Update", Evaluate(function="Update_Preview"), background=lightorange),
Button[ClosePreview]("Close", CloseWindow(PREVIEW), background=pink)
],
[ MathMLViewer[MMLPreview](value=MathML[Export](""), width=500, height=100)
]
]),
Action[APreM](Evaluate(function="Preview(MInt)")),
Action[APreMyz](Evaluate(function="Preview(MyzInt)")),
Action[APreMxz](Evaluate(function="Preview(MxzInt)")),
Action[APreMxy](Evaluate(function="Preview(MxyInt)")),
Action[ARunPreview](RunWindow(PREVIEW))
):Maplets[Display](CenterOfMass);