(*
CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265 <##### NEW ADDRESS ####

Monday, September  25:  Section 13.8

p979:5,7,11,19

Previous Assignment.
p973:29,31,35

No hand-in-assignment.  We worked the problem as

an inclass group-work.

*)

(* This is a mathematica assignment to study flotation *)
(* The Pressure acts perpendicular to the surface and  *)
(* is proportional to the depth.  Find the Center of   *)
(* Pressure for a cone z = Sqrt[x^2+y^2] and the       *)
(* paraboloid z = x^2 + y^2 up to a depth of z = 10.    *)

(*
u[{x_,y_,z_}] := 1.0/Sqrt[x^2+y^2+z^2] {x,y,z};    (* Makes unit vector *)

c[{x_,y_}] := Sqrt[x^2+y^2];                    (* Equation of the cone *)

p[{x_,y_}] := x^2+y^2;                    (* Equation of the paraboloid *)

normal[{x_,y_}] := 1.0/Sqrt[2x^2 + 2 y^2] {-x,-y,Sqrt[x^2+y^2]}  (* Unit normal for cone *)

pt[n_,i_,j_] := { 10.0 i/n Cos[2.0 Pi j/n], 10.0 i/n Sin[2.0 Pi j/n]};

dS[n_,i_,j_] := Sqrt[2.0] 10.0 i/n (10.0/n)(2.0 Pi/n);   (* Patch of Surface Area *)

g[n_,i_] := Sum[ normal[pt[n,i,j]] (10.0-c[pt[n,i,j]]) dS[n,i,j] ,{j,1,n}];

f[n_] := Sum[ g[n,i],{i,1,n}]; 

*)

(* We take the cone z = r from z=0 to z=h and place it below    *)

(* the surface of the water.   The surface of the water will be *)

(* the place  ax + by + cz + d = 0.  The "depth" of a point (x,y,z) *)

(* will be the distance from the plane.                             *)

(* depth = (ax+by+cz+d)/Sqrt[a^2+b^2+c^2]                           *)


(* The total bouyancy is                                            *)


(* Int[ depth(x,y,z) unitnormal(x,y,z) dS, over the base of cone]   *)

(* + Int[ depth(x,y,h) (0,0,1) dA over the base of the cone]        *)

a = 1; b = 3; c = 6; d = 2000;

h = 5; 


depth[x_,y_,z_] := (a x + b y + c z + d)/Sqrt[a^2+b^2+c^2];


(* depth[x_,y_,z_] :=  d-z; *)

(* unitnormal[x_,y_] := 1/Sqrt[x^2+y^2+1] {-x,-y,+1}; *)

unitnormal[r_,t_] := 1/Sqrt[2] {-Cos[t],-Sin[t],1}; 

f1 = Integrate[ depth[r Cos[t], r Sin[t], r] unitnormal[r,t] Sqrt[2] r,
{r,0,h},{t,0,2 Pi}];


f2 = Integrate[depth[r Cos[t], r Sin[t], h] {0,0,-1} r, {r,0,h}, {t, 0, 2 Pi}];

Print[f1+f2,"      ",Sqrt[(f1+f2).(f1+f2)]];

(* moment about the axis  {p,q,s} through {xo,yo,zo}   *)

xo = 0; yo = 0; zo = 3/4 h;

p = 0; q = 1; s = 3;

m[r_,t_] := Det[ { unitnormal[r,t], {xo-r Cos[t], yo-r Sin[t], zo-r}, {p,q,s} } ]; 

m1 = Integrate[ depth[r Cos[t], r Sin[t], r] m[r,t] Sqrt[2] r,
{r,0,h},{t,0,2 Pi}];

m2 = Integrate[depth[r Cos[t], r Sin[t],h]* 
Det[{{0,0,-1},{xo-r Cos[t],yo-r Sin[t],h-r},{p,q,r}}] r,{r,0,h},{t,0, 2 Pi}];

Print[" Total Moment is ",m1+m2];

volume = Integrate[ r,{t,0,2 Pi},{r,0,h},{z,r,h}];

moment = Integrate[ z r,{t,0,2 Pi},{r,0,h},{z,r,h}];

zbar = moment/volume;

Print["z bar is ",zbar];