Problem #1)

ANSWERS : 
M1: Yes, rotation by θ 
M2:   No, need one negative and one positive Sinθ term
M3:  Yes, rotation by 30 Degrees
M4:  Yes, rotation by 180 Degrees
M5:  Yes 3-D rotation about the x-axis  by θ 
M6:  No, this flips all axes. (This is a parity transformation.)
M7:  No, this will stretch the axes. 
M8:  No, this will stretch x-axis and compress y-axis

Problem #2)

Clear["Global`*"]

m=(({{4, 0, 0}, {0, (11/2), (3/2)}, {0, (3/2), (11/2)}}));

m2=(({{(11/2), (3/2)}, {(3/2), (11/2)}}));

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

Det[m - λ one]

sol=Solve[Det[m - λ one]==0,λ]

Eigenvalues[m]

Eigenvalues[m2]

eqs= (m -λ one).{a,b,c} ==0 //Thread

sol[[1]]

sol1=Solve[ eqs /.sol[[1]],{a,b,c}][[1]]

v1={a,b,c} /.sol1

v1={a,b,c} /.sol1 /.{a->1,b->0}

sol[[3]]

sol3=Solve[ eqs /.sol[[3]],{a,b,c}][[1]]

v3={a,b,c} /.sol3

v3={a,b,c} /.sol3 /.{b->1}

v2=Cross[v1,v3]

{a,b,c} /.sol1 /.{a->0,b->-1}

{v1,v2,v3}

Problem #3)

Clear["Global`*"]

m={{((Sqrt[3])/2),0,1/2},{1/2,0,-((Sqrt[3])/2)},{0,1,0}};
one=DiagonalMatrix[{1,1,1}];
m//MatrixForm

m.Transpose[m]

Eigenvalues[m]

eqs=(m-λ one).{a,b,c} ==0 /.{λ->1} //Thread

sol=Solve[eqs,{a,b,c}][[1]]

vec1={a,b,c} /.sol /.{a->1}

(vec1/(2-(Sqrt[3])))

(vec1/(2-(Sqrt[3]))) //Simplify

vec1 //Normalize //Simplify

eq=Tr[m]==1+2 Cos[θ]

Solve[eq,Cos[θ]][[1]]

tsol=Solve[eq,θ][[1]]

tsol1=Solve[eq,θ][[1]]  /.{C[_]:>0}

tsol2=Solve[eq,θ][[1]]  /.{C[_]:>1}

θ0=(θ/Degree) /.tsol1 //N

θ0=(θ/Degree) /.tsol2 //N

1/(2 - (Sqrt[3])) //Simplify

m.{1/(2 - (Sqrt[3])),1,1} //Simplify

m.{(2 + (Sqrt[3])),1,1} //Simplify

Problem #4)

Clear["Global`*"]

Tmat=m  DiagonalMatrix[{1,1,1}];
Tmat //MatrixForm

Vmat=k{{2,1,-1},{1,2,1},{-1,1,2}};
Vmat //MatrixForm

mat= Vmat-Tmat ω2;
mat //MatrixForm

Det[mat] //Factor

sol=Solve[Det[mat] ==0,ω2] //Simplify

eq1=mat .{a,b,c} ==0 //Thread;
eq1 //Column

sol1=Solve[eq1 /.sol[[1]],{a,b,c}][[1]]   //Simplify

ev1={a,b,c} /.sol1 /.{a->1}//Simplify

sol2=Solve[eq1 /.sol[[2]],{a,b,c}][[1]]   //Simplify //PowerExpand

ev2={a,b,c} /.sol2 /.{a->1} //Simplify

ev2=ev2 /.{b->1}

ev3=Cross[ev1,ev2]

eVecs=Normalize /@ {ev1,ev2,ev3} //Simplify

Vdiag=eVecs.Vmat.Transpose[eVecs]  //FullSimplify ;
Vdiag//MatrixForm

Tdiag=eVecs.Tmat.Transpose[eVecs]  //FullSimplify ;
Tdiag //MatrixForm

Part B Look at motion

values={m->1,k->1};

sol /.values

mode1=ev1 Exp[I ω t] /.{ω->Sqrt[ω2]} /.sol[[1]]  /.values

mode1=ev1 (x0 + v0 t ) /.{ω->Sqrt[ω2]} /.sol[[1]]  /.values /.{x0->0,v0->1}

mode2=ev2 Exp[I ω t] /.{ω->Sqrt[ω2]} /.sol[[2]]/.values

mode3=ev3 Exp[I ω t] /.{ω->Sqrt[ω2]} /.sol[[3]]/.values

mode1 +{-1,1} /.values

Plot[mode1 +{-1,0,1}/.values //Re,{t,0,10}]

Plot[mode2 +{-2,0,2}/.values //Re,{t,0,10}]

Plot[mode3 +{-2,0,2}/.values //Re,{t,0,10}]

Problem #5)

Clear["Global`*"]

T=(1/2)(2m) ((x1'[t])^2)+(1/2)(1m) ((x2'[t])^2);

V=(1/2)(2k) ((x1[t])^2)+(1/2)(1k) ((x2[t])^2) + (1/2)(2k )(((x1[t]-x2[t]))^2);

lag=T-V

D[D[lag,x1'[t]],t] - D[lag,x1[t]] //Simplify

D[D[lag,x2'[t]],t] - D[lag,x2[t]] //Simplify

Tmat=m  DiagonalMatrix[{2,1}];
Tmat //MatrixForm

Vmat=k{{4,-2},{-2,3}};
Vmat //MatrixForm

mat= Vmat-Tmat ω2;
mat //MatrixForm

sol=Solve[Det[mat] ==0,ω2] //Simplify

eq1=mat .{a,b} ==0 //Thread;
eq1 //Column

sol1=Solve[eq1 /.sol[[1]],{a,b}][[1]]   //Simplify

ev1={a,b} /.sol1 /.{a->1}//Simplify

sol2=Solve[eq1 /.sol[[2]],{a,b}][[1]]   //Simplify //PowerExpand

ev2={a,b} /.sol2 /.{a->1} //Simplify

eVecs=Normalize /@ {ev1,ev2} //Simplify

Vdiag=eVecs.Vmat.Transpose[eVecs]  //FullSimplify ;
Vdiag//MatrixForm

Tdiag=eVecs.Tmat.Transpose[eVecs]  //FullSimplify ;
Tdiag //MatrixForm

Problem #6)

Clear["Global`*"]

T=(1/2)m1 ((x'[t])^2)+(1/2)(inertia) ((w'[t])^2);

V=(1/2)k ((x[t])^2)-m1 g x[t];

inertia = (1/2)m2 (r^2);
w'[t] = ((x'[t])/r);

T

V

lag=T-V

eqP= p[t] == D[lag,x'[t]] //Factor

psol=DSolve[ eqP,p[t],t][[1]]

xsol=Solve[ eqP,x'[t]][[1]]

ham=T+V

ham= ham /.xsol //Simplify

eq1= D[ham,p[t]] ==x'[t] //Simplify

eq2=- D[ham,x[t]] ==p'[t]

p2sol=DSolve[eq1,p,t][[1]]

eqHam=eq2 /.p2sol //Simplify

Check that Lagrange equations are equivalent

D[D[lag,x'[t]],t] - D[lag,x[t]] //Simplify

(* Same thing, just multiply by 2 and move to other side:*)
eqLag=2D[D[lag,x'[t]],t] ==2 D[lag,x[t]] //Simplify

eqLag == eqHam

Problem #7)

Problem A

Clear["Global`*"]

ntar= ((ρ t)/m) /.{ρ->(11.34 (10^3)), t->(0.3 (10^(-3))), m->(208  *(1.66 (10^(-27))))}

Nsc = Nin ntar sig  /.{Nin-> (10^4),sig->50 (10^(-28))}

Problem B

Clear["Global`*"]

solidAngle= Integrate[Sin[θ], {θ,0,π},{ϕ,0,2π}]

coverage= Integrate[Sin[θ], {θ,θ1,θ2},{ϕ,0,2π}]

coverage /.{θ1 -> 10 Degree,θ2 -> 20 Degree}

ratio=(coverage/solidAngle)/.{θ1 -> 10 Degree,θ2 -> 20 Degree}

ratio  //N

(10^(6 ))ratio  //N

Problem C

Clear["Global`*"]

θlab=60 Degree

θcms=2 θlab