Spherical Cone

Author

Eric W. Weisstein
December 27, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Volume

Integrate[Boole[r<R](r^2)Sin[ϕ],{r,0,∞},{θ,0,2π},{ϕ,0,ϕ},Assumptions->R>0]

Sin[ϕ]==(r/R)

r==(Sqrt[h(2R-h)])

v=FullSimplify[(4/3) π (R^3) ((Sin[(ϕ/2)])^2)/.ϕ:>ArcSin[((Sqrt[h(2R-h)])/R)],0<h<R]

Centroid

zbar=Integrate[r Cos[ϕ]Boole[r<R](r^2)Sin[ϕ],{r,0,∞},{θ,0,2π},{ϕ,0,ϕ},Assumptions->R>0]

FullSimplify[(zbar/v)/.ϕ:>ArcSin[((Sqrt[h(2R-h)])/R)],0<h<R]

Invertia Tensor

With[{ρ=(M/((4/3)π (a^3)))},ρ Integrate[Boole[r<a]InertiaTensor[{r,θ,ϕ},"CoordinateSystem"->"Spherical"](r^2)Sin[ϕ],{r,0,∞},{θ,0,2π},{ϕ,0,2π},Assumptions->a>0]
]

i=With[{ρ=(M/v)},ρ Integrate[Boole[r<R]InertiaTensor[{r,θ,ϕ},"CoordinateSystem"->"Spherical"](r^2)Sin[ϕ],{r,0,∞},{θ,0,2π},{ϕ,0,ϕ},Assumptions->R>0]
]

FullSimplify[i/.ϕ:>ArcSin[((Sqrt[h(2R-h)])/R)],0<h<R]

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