m=1;  l=1;g=1;k=0.1

eqs= {(l+a[t])*b''[t]+2*b'[t]*a'[t]+g*Sin[b[t]]==0,a''[t]==-k/m*a[t]}

condinit={a[0]==0.1,b[0]==Pi/2.0,a'[0]==0,b'[0]==0.1}

solution=NDSolve[{eqs,condinit},{a,b},{t,0,1000}]

Plot[Evaluate[{a[t],b[t]}/.solution],{t,0,100}]

c1[t_]=Evaluate[{(l+a[t])*Sin[b[t]],-1.0*(l+a[t])*Cos[b[t]]}/.solution]

Plot[c1[t],{t,0,100}]

ParametricPlot[{c1[t]},{t,0,100},PlotRange->{{-1.3,1.3},{-1.25,0}}]

Animate[Graphics[{{PointSize[.025],{Blue,Point[c1[t]]},Line[{{0,0},Flatten[c1[t]]}]},{Thickness[0.0002],{Orange,Line[Map[Function[c1[#]],Range[0,t,0.01]]]}}},PlotRange->{{-1.3,1.3},{-1.25,0.5}},Axes->True,Ticks->False,ImageSize->300],{t,0,100,0.01},AnimationRate->1.,SaveDefinitions->True]