(* Contributed by Noah Rubin, 2015-01-20 *)
R=5;
g=9.8;
α=30 Degree; (**INITIAL CONDITION: ENTER HERE**)
s=NDSolve[{ϕ''[t]==-(g/R)Sin[ϕ[t]],ϕ[0]== α,ϕ'[0]==0},ϕ,{t,0,100}]
x[a_]:=R Sin[ϕ[a]/.s];
y[a_]:=R-R Cos[ϕ[a]/.s];


θ[a_]:=α Cos[(Sqrt[(g/R)])a];
w[a_]:=R Sin[α Cos[(Sqrt[(g/R)])a]];
h[a_]:=R - R Cos[α Cos[(Sqrt[(g/R)])a]];

Animate[
GraphicsColumn[{Graphics[{Orange,Disk[Flatten@{x[t],y[t]},0.5],Thick, Line[{{0,R},Flatten@{x[t],y[t]}}],Blue,Disk[{w[t],h[t]},0.25],Line[{{0,R},{w[t],h[t]}}]},PlotRange->{{-5,5},{-1,5}}],

Plot[ϕ[q]/.s,{q,0,t},PlotStyle->Orange,AxesLabel->{"t (s.)","ϕ (rad.)"},PlotLabel->"Numerical Solution"],Plot[θ[q],{q,0,t},PlotStyle->Blue,AxesLabel->{"t (s.)","ϕ (rad.)"},PlotLabel->"Small Angle Approximation"]}]


,{t,1,15,0.005},AnimationRate->1,RefreshRate->75]