Cylindrical Hoof

Author

Eric W. Weisstein
April 4, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/CylindricalHoof.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CylindricalHoof.html.

©2013 Wolfram Research, Inc. except for portions noted otherwise

Cross Section

{x1,y1,z1}={0,-r,0};
{x2,y2,z2}={0,r,0};
{x3,y3,z3}={-r,0,h};
Factor[Det[{{x,y,z,1},{x1,y1,z1,1},{x2,y2,z2,1},{x3,y3,z3,1}}]]

ParametricPlot3D[Evaluate[{{x,Sqrt[r^2-x^2],h x/r},{x,-Sqrt[r^2-x^2],h x/r}}/.{h->5,r->1}],{x,0,r},Ticks->None]

Rx[t_]:={{1,0,0},{0,Cos[t],Sin[t]},{0,-Sin[t],Cos[t]}}
Rz[t_]:={{Cos[t],Sin[t],0},{-Sin[t],Cos[t],0},{0,0,1}}

Simplify[Rz[ArcTan[h/r]].(Rx[Pi/2].{x,Sqrt[r^2-x^2],h x/r})]

N[Sqrt[r^2+h^2]]

h=5;r=1;
ParametricPlot[{
   {x/r Sqrt[r^2+h^2],Sqrt[r^2-x^2]},
   {x/r Sqrt[r^2+h^2],-Sqrt[r^2-x^2]}
 },{x,0,r},
 AspectRatio->Automatic,Ticks->None]

h=.;
x[t_]:=t/r Sqrt[r^2+h^2]
y[t_]:=Sqrt[r^2-t^2]
dsdt[t_]:=Sqrt[Simplify[x'[t]^2+y'[t]^2]]

Plot[Evaluate[dsdt[t] /. {h->5,r->1}],{t,0,1}]

dsdt[t]

Integrate[dsdt[t],{t,0,r}]

Centroid

FullSimplify[Integrate[1,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

xbar=FullSimplify[Integrate[x ,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

PowerExpand[Simplify[xbar /. b->a]]

xbar=FullSimplify[Integrate[x ,{x,0,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h x)/a)}]/V[h,a,a],a>0]

ybar=FullSimplify[Integrate[y,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

zbar=FullSimplify[Integrate[z ,{x,a-b,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,0,((h (-a+b+x))/b)}]/V[h,a,b],a>0&&b>0]

PowerExpand[Simplify[zbar /. b->a]]