1) 9.8 Coriolis and Centrifugal Forces

Clear["Global`*"]

a) moving south near north pole

(* a: moving south near north pole *)
Ω = Ω0{0,0,1};
r={0,Sin[θ],Cos[θ]};
v=v0{0,+Cos[θ],-Sin[θ]};

(* Direction: west: +x direction *)
coriolis=2 m Cross[v,Ω]

(* Direction: up and south: +y direction *)
centrifugal= m Cross[Cross[Ω,r],Ω]

b) moving east near equator

(* a: moving south near north pole *)
Ω = Ω0{0,0,1};
r={0,Sin[θ],Cos[θ]};
v=v0{-1,0,0};

(* Direction: up: +y direction *)
coriolis=2 m Cross[v,Ω]

(* Direction: up: +y direction *)
centrifugal= m Cross[Cross[Ω,r],Ω]

centrifugal/.{θ->π/2}

c) moving south near equator

(* a: moving south near north pole *)
Ω = Ω0{0,0,1};
r={0,Sin[θ],Cos[θ]};
v=v0{0,0,-1};

(* Direction: up: +y direction *)
coriolis=2 m Cross[v,Ω]

(* Direction: up: +y direction *)
centrifugal= m Cross[Cross[Ω,r],Ω]

centrifugal/.{θ->π/2}

2) 9.9 Bullet problem

Clear["Global`*"]

Ω = Ω0{0,0,1};
r={0,Sin[θ],Cos[θ]};
v=v0{0,-Cos[θ],Sin[θ]};

(* Direction:east:-x direction *)
coriolis= 2 m Cross[v,Ω]

values= {θ->40 Degree, v0->1000, Ω0->(( 2π)/86400),g->9.8};

tmp1=coriolis[[1]] /.values

(tmp1/(m g)) //.values

3) 9.13 Plumb line problem

Clear["Global`*"]

Ω = Ω0{0,0,1};
r=re{0,Sin[θ],Cos[θ]};

centrifugal= m Cross[Cross[Ω,r],Ω]

(* Take part perpendicular to gravity:Cos[θ] *)
tmp1= centrifugal[[2]] Cos[θ]

tanα= (tmp1/(m g))

tanα //TrigReduce

5) Foucault Pendulum

Clear["Global`*"]

Ωz = Ω Cos[θ]

eta = amp Exp[-I Ωz t ] Cos[ω0 t]

values={ω0-> 16 Ω, amp->1, Ω->2π}

f[t_,θ_]=eta //.values

Plot[ {f[t,π/3] //Re,f[t,π/3] //Im},{t,0,2}]

titles={"North Pole","Dallas","Equator","South America","South Pole"};

ff[x_]:=ParametricPlot[ {f[t,x (π/4)] //Re,f[t,x (π/4)] //Im} ,
{t,0,1/4},
PlotLabel->{titles[[x+1]],x," θ= ",x (π/4)}]

Table[ff[i],{i,0,4,1}] //TableForm

Problem 6

Clear["Global`*"]

m={{Cos[θ],Sin[θ]},{-Sin[θ],Cos[θ]}};
m //MatrixForm

one=DiagonalMatrix[{1,1}];
one //MatrixForm

m - λ one //MatrixForm

eq= Det[m - λ one] //Simplify

sol=Solve[eq==0,λ]

sol //TrigToExp

λ1= Exp[+I θ] //ExpToTrig

λ2= Exp[-I θ] //ExpToTrig

vec={a,b}

eqs=m.vec == λ1 vec

Solve[eqs,{a,b}]

Solve[m.vec == λ2 vec,{a,b}]

vec1={1,I};
vec2={1,-I};

m.vec1 == λ1 vec1 //Simplify

m.vec2 == λ2 vec2 //Simplify

Eigensystem[m] //MatrixForm

Problem 7

Clear["Global`*"]

m={{Cos[θ],Sin[θ],0},{-Sin[θ],Cos[θ],0},{0,0,1}};
m //MatrixForm

one=DiagonalMatrix[{1,1,1}];
one //MatrixForm

m - λ one //MatrixForm

eq= Det[m - λ one] //Simplify

sol=Solve[eq==0,λ]

sol //TrigToExp

{λ3,λ2,λ1}=λ /.sol

vec={a,b,c};
eq= m.vec == λ vec

Solve[eq /.sol[[1]],{a,b,c}]

Solve[eq /.sol[[2]],{a,b,c}]

Solve[eq /.sol[[3]],{a,b,c}]

vec1={1,I,0};
vec2={1,-I,0};
vec3={0,0,1};

m.vec1 == λ1 vec1 //Simplify

m.vec2 == λ2 vec2 //Simplify

m.vec3 == λ3 vec3 //Simplify

Eigensystem[m] //MatrixForm

Problem 8

Clear["Global`*"]

m={{5,-(Sqrt[3])},{-(Sqrt[3]),7}}/4;
m //MatrixForm

one=DiagonalMatrix[{1,1}];
one //MatrixForm

m - λ one //MatrixForm

eq= Det[m - λ one] //Simplify

sol=Solve[eq==0,λ]

sol //TrigToExp

{λ1,λ2}=λ /.sol

vec={a,b};
eq= m.vec == λ vec

Solve[eq /.sol[[1]],{a,b}]

Solve[eq /.sol[[2]],{a,b}]

vec1={1,(1/(Sqrt[3]))};
vec2={1,-(Sqrt[3])};

m.vec1 == λ1 vec1 //Simplify

m.vec2 == λ2 vec2 //Simplify

Eigensystem[m] //MatrixForm

{(vec2/(-(Sqrt[3]))),(Sqrt[3])vec1}