Horizontal Cylindrical Segment

Author

Eric W. Weisstein
June 10, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

With[{a=1,L=1,h=.2},
ParametricPlot3D[{v,a Cos[u],a Sin[u]},{u,-Pi-ArcCos[(a-h)/"a"],-Pi+ArcCos[("a"-"h")/"a"]},{"v",0,"L"}]
]

With[{h=.2,a=1,L=2},RegionPlot3D[y^2+z^2<a^2&&z<h&&0<x<L,{x,0,2},{y,-2,2},{z,-1,1},Boxed->False,BoxRatios->Automatic,Axes->False,PlotPoints->50]]

Volume

v[L_,r_,h_]:=L r^2 ArcCos[(r-h)/r]-(r-h)L Sqrt[2r h-h^2]

Assuming[L>0&&a>0&&0<h<a,Volume[y^2+z^2<a^2&&z<-(a-h)&&0<x<L,{x,y,z}]]//Timing

FullSimplify[%,L>0&&a>0&&0<2h<a]

Assuming[L>0&&a>0&&0<h<2a,Volume[y^2+z^2<a^2&&z<h-a&&0<x<L,{x,y,z}]]//Timing

FullSimplify[Last[%54],L>0&&a>0&&0<h<2a&&h>a]

v2[L_,a_,h_]:=Piecewise[{
{L ((Sqrt[(2 a-h) h]) (-a+h)+(a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))]),h≤a},
{L ((Sqrt[(2 a-h) h]) (-a+h)-(a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))])),h>a}
}]

FullSimplify[(ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))]),a>0&&0<h<2a]

FullSimplify[L ((Sqrt[(2 a-h) h]) (-a+h)-(a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))]))-L ((Sqrt[(2 a-h) h]) (-a+h)+(a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))]),a>0&&0>h<2a&&L>0]

With[{L=2,a=1},Plot[L ((Sqrt[(2 a-h) h]) (-a+h)-(a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))]))-L ((Sqrt[(2 a-h) h]) (-a+h)+(a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))]),{h,0,2a}]]

h==0

V[L,r,0]

Limit[v2[L,a,h],h->0]

h==r

V[L,r,r]

Simplify[Limit[v2[L,a,h],h->a],a>0]

h==2 r

V[L,r,2r]

<<Utilities`Typesetting`

Plot[V[1,1,h],{h,0,2},PlotStyle->Red,AxesLabel->{InlineFraction[h,r],InlineFraction[V,L]}]

Plot[v2[1,1,h],{h,0,2},PlotStyle->Red]

Attempt toapproximate

FullSimplify[V[L,r,h]/.h->r/x,r>0]

Solve[(( (-(-1+x) (Sqrt[((-1+2 x)/(x^2))])+x ArcCos[((-1+x)/x)]))/x)==V,x]

Simplify[Normal[Series[V[L,r,h],{h,0,2},Assumptions->h>0&&r>0]],r>0]

FullSimplify[Solve[(4/3) (Sqrt[2]) (h^(3/2)) L (Sqrt[r])==V,h],r>0]

Simplify[Normal[Series[V[L,r,h],{h,0,6},Assumptions->h>0&&r>0]],r>0]

With[{L=1,r=1},
Plot[Evaluate[{V[L,r,h],(4/3) (Sqrt[2]) (h^(3/2)) L (Sqrt[r]),-(((h^(3/2)) L (1575 (h^4)+6160 (h^3) r+31680 (h^2) (r^2)+354816 h (r^3)-2365440 (r^4)))/(887040 (Sqrt[2]) (r^(7/2))))}],{h,0,2},PlotStyle->{Red,Blue,Green}]
]

Surface Area

area[a_,L_,h_]:=2L ArcCos[(a-h)/a]

area[a,L,a]

area[a,L,2a]

area[a,L,0]

Centroid

Assuming[L>0&&a>0&&0<h<a,Centroid[y^2+z^2<a^2&&z<-(a-h)&&0<x<L,{x,y,z}]]//Timing

FullSimplify[{(L/2),0,((2 ((((2 a-h) h))^(3/2)))/(3 (a (Sqrt[(2 a-h) h])-h (Sqrt[(2 a-h) h])-(a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))])))},L>0&&a>0&&0<h<a]

Assuming[L>0&&a>0&&a<h<2a,Centroid[y^2+z^2<a^2&&z<h-a&&0<x<L,{x,y,z}]]//Timing

FullSimplify[Last[%93],L>0&&a>0&&a<h<2a&&h>a]

Assuming[L>0&&a>0&&a<h<2a&&h>0,Centroid[y^2+z^2<a^2&&z<h-a&&0<x<L,{x,y,z},GenerateConditions->True]]//Timing

FullSimplify[Last[%93],L>0&&a>0&&a<h<2a&&h>a]

centroid[a_,h_,L_]:=Piecewise[{
{{(L/2),0,((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])-3 (a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))]))},h<=a},
{{(L/2),0,((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])+3 (a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))])))},h>a}
}]

Simplify[Limit[{((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])-3 (a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))])),((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])+3 (a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))])))},h->a,Direction->#]&/@{-1,1},a>0]

Plot[centroid[1,h,2][[3]],{h,0,2}]

With[{a=1,L=2},Plot[(((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])-3 (a^2) ArcCsc[(a/(Sqrt[(2 a-h) h]))]))/((2 ((((2 a-h) h))^(3/2)))/(3 (a-h) (Sqrt[(2 a-h) h])+3 (a^2) (ArcCot[((a-h)/(Sqrt[(2 a-h) h]))]-2 ArcSec[(a/(Sqrt[(2 a-h) h]))])))),{h,0,2}]]

IntertiaTensor

i=With[{ρ=(M/(L a^2 ArcCos[(a-h)/a]-(a-h)L Sqrt[2a h-h^2]))},
ρ Integrate[InertiaTensor[{x,y,z},"CoordinateSystem"->"Cartesian"],
{x,0,L},{z,-a,-(a-h)},{y,-(Sqrt[(a^2)-(z^2)]),(Sqrt[(a^2)-(z^2)])},Assumptions->0<h<a&&a>0&&h>0]
];//Timing

i

FullSimplify[i/.(Sqrt[(2 a-h) (h^(k_))])->(h^(Floor[(k-1)/2]))(Sqrt[(2 a-h) h])]

i=With[{ρ=(M/(L a^2 ArcCos[(a-h)/a]-(a-h)L Sqrt[2a h-h^2]))},
ρ Integrate[InertiaTensor[{x,y,z},"CoordinateSystem"->"Cartesian"]Boole[y^2+z^2<a^2&&z<h-a],
{x,0,L},{y,-∞,∞},{z,-∞,∞},Assumptions->0<a<h<2a&&a>0,GenerateConditions->True]
];//Timing

i=With[{ρ=(M/(L a^2 ArcCos[(a-h)/a]-(a-h)L Sqrt[2a h-h^2]))},
ρ Integrate[InertiaTensor[{x,y,z},"CoordinateSystem"->"Cartesian"],
{x,0,L},{z,0,h-a},{y,-(Sqrt[(a^2)-(z^2)]),(Sqrt[(a^2)-(z^2)])},Assumptions->a<h<2a&&a>0&&h>0]
];//Timing

i

FullSimplify[i/.(Sqrt[(2 a-h) (h^(k_))])->(h^(Floor[(k-1)/2]))(Sqrt[(2 a-h) h])]

Vector Properties

eqn={v,a Cos[u],a Sin[u]};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&(u|v)∈Reals;

Vector Length

Assuming[assum,FullSimplify[Norm[eqn]/.rules]]

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{u,v}]/.rules]]