{Slider[Dynamic[r0],{0,2,0.1}],Dynamic[r0]}

rdot=0.05

Animate[Graphics[{Blue,Circle[{0.0,0.0},r0],Black, Line[{{-1.2,0},{1.2,0}}],Line[{{0,-1.2},{0,1.2}}],Red,Disk[{r0*Sin[thet],-r0*Cos[thet]},rdot],Black,Inset[Style[NumberForm[thet,{3,2}],20],{1.0,1.1}]}],{thet,0,25},AnimationRunning->False,AnimationRepetitions->1]

g=9.81

theta0=0.2

thetadot0=0

Interesting values: 1.0, 4.0, 6.27,7

{Slider[Dynamic[omega],{0,20,0.5}],Style[NumberForm[Dynamic[omega],{3,2}]]}

s=NDSolve[{theta''[t]==Sin[theta[t]]*Cos[theta[t]]*omega^2-g/r0*Sin[theta[t]],theta[0]==theta0,theta'[0]==thetadot0},theta,{t,0,100,0.001}]

{Slider[Dynamic[anrate],{0.1,1,0.05}],Style[NumberForm[Dynamic[anrate],{3,2}]]}

Animate[Graphics[{Blue,Circle[{0.0,0.0},r0],Black, Line[{{-1.2,0},{1.2,0}}],Line[{{0,-1.2},{0,1.2}}],Black,Inset[Style[NumberForm[t,{3,2}],20],{1.0,1.1}],Red,Disk[{r0*Sin[theta[t]],-r0*Cos[theta[t]]},rdot]}/.s],{t,0,100,0.001},AnimationRunning->False,AnimationRepetitions->1,AnimationRate->Dynamic[anrate]]

Table[r0*g*i,{i,1,5}]

Dynamic[r0*thetadot0^2]

Dynamic[ArcCos[g/r0/omega^2]]

Dynamic[2*Pi/omega/Sin[theta0]]

Dynamic[2*Pi/Sqrt[g/r0-omega^2]]