Dear Ă–mer,
I don't know what you really want to do, what the physical situation is you want to describe and if your equations are correct.
Anyhow, what about this? Is it this what you want to see?
z1 = {-R*W*Sin[W*t] + l'[t]*Sin[\[Phi][t]] +
l[t]*(\[Phi]'[t])*Cos[\[Phi][t]],
R*W*Cos[W*t] - l'[t]*Cos[\[Phi][t]] +
l[t]*(\[Phi]'[t])*Sin[\[Phi][t]]};
V = m*g*(R*Sin[W*t] - l[t]*Cos[\[Phi][t]]) + 1/2*k*(l[t] - l0)^2;
T = 1/2*m*z1.z1;
Lagrange = T - V;
eqs = D[D[Lagrange, \[Phi]'[t]], t] - D[Lagrange, \[Phi][t]];
eqs2 = D[D[Lagrange, l'[t]], t] - D[Lagrange, l[t]];
g = 9.7; m = 1; l0 = 1; k = 1000; R = 2; W = Pi/2;
sol = NDSolve[{eqs == 0, eqs2 == 0, l[0] == l0, l'[0] == 0,
Derivative[1][\[Phi]][0] == 0, \[Phi][0] == 0}, {l[t], \[Phi][
t]}, {t, 0, 20}];
p1 = R {Cos[W t], Sin[W t]};
z1a = z1 /. Flatten[sol /. a_[t] -> a];
Animate[Graphics[{Circle[{0, 0}, R], PointSize[.05], Black,
Point[p1 /. t -> tt], Red, Point[z1a /. t -> tt], Blue,
Line[{p1, z1a} /. t -> tt]}, Axes -> True,
PlotRange -> {{-4, 4}, {-4, 4}}], {tt, 0, 20}]