m1=100; l1=10; x1=0.5; v1=0.5;
m2=1; l2=1; x2=1.0; v2=0.0;
g=9.81;
Result=NDSolve[
{p1[0]==x1,p1'[0]==v1,p2[0]==x2,p2'[0]==v2,
p1''[t]==-(g/l1)Sin[p1[t]]-(m2/(m1+m2))(l2/l1)(Cos[p2[t]-p1[t]]* p2''[t]+Sin[p1[t]-p2[t]](p2'[t])^2),p2''[t]==-(g/l2)Sin[p2[t]]-(l1/l2)(Cos[p2[t]-p1[t]] p1''[t]-Sin[p1[t]-p2[t]] (p1'[t])^2)},
{p1,p2},{t,0,20},
MaxSteps->500000]
Animate[
Graphics[
{Disk[{l1*Sin[p1[t]],-l1*Cos[p1[t]]}, 0.3], Text["m1",{l1*Sin[p1[t]]+1.0,-l1*Cos[p1[t]]}],
Disk[{l1*Sin[p1[t]]+l2*Sin[p2[t]],-l1*Cos[p1[t]]-l2*Cos[p2[t]]}, 0.1], Text["m2",{l1*Sin[p1[t]]+l2*Sin[p2[t]]+0.8,-l1*Cos[p1[t]]-l2*Cos[p2[t]]}],
Line[{{0,0},{l1*Sin[p1[t]],-l1*Cos[p1[t]]}, {l1*Sin[p1[t]]+l2*Sin[p2[t]],-l1*Cos[p1[t]]-l2*Cos[p2[t]]}}]},
PlotRange->12.25],
{t,0,13,0.02}]/.Result