s1[t_] := R1 \[Phi]1[t];
x1[t_] := x10 + s1[t] Cos[\[Alpha]] + R1 Sin[\[Alpha]];
y10 = Abs[x10] Tan[\[Alpha]];
y1[t_] := y10 - s1[t] Sin[\[Alpha]] + R1 Cos[\[Alpha]];
T1[t_] := 1/2 m1 (D[x1[t], t]^2 + D[y1[t], t]^2) + 1/2 I1 (D[\[Phi]1[t], t])^2;
V1[t_] := m1 g y1[t];
L = FullSimplify[T1[t] - V1[t]]
eq1 = FullSimplify[D[D[L, \[Phi]1'[t]], t] - D[L, \[Phi]1[t]]]

m1 = 1; R1 = 1; g = 1; \[Alpha] = \[Pi]/6; x10 = -20;
sol = Table[
   NDSolve[{(eq1 == 0) /. {I1 -> j}, \[Phi]1[0] == 0, \[Phi]1'[0] == 1}, {\[Phi]1[t]}, {t, 0, 20}], {j, 0, 5}];
dim = Dimensions[sol][[1]];
\[CapitalDelta] = 2;
p1 = Table[ Legended[Graphics3D[{
Gray, Polygon[{{0, \[CapitalDelta], 0}, {1.2*x10, \[CapitalDelta], 1.2 y10}, {1.2*x10, (dim + 1) \[CapitalDelta], 1.2 y10}, {0, (dim + 1) \[CapitalDelta], 0}}], Table[{Hue[j/dim], Cylinder[({{x1[t], \[CapitalDelta] j, y1[t]}, {x1[t], \[CapitalDelta] (j + 1), y1[t]}} /. sol[[j]]) /. {t -> \[Tau]}, R1]}, {j, 1, dim}]}, Boxed -> False],
    PointLegend[Table[Hue[j/dim], {j, 1, dim}] , Table[StringJoin["m=\!\(\*SubscriptBox[\(m\), \(0\)]\)   I=", ToString[j], "\!\(\*SubscriptBox[\(I\), \(0\)]\)"], {j, 1, dim}], LegendMarkerSize -> 20,      LabelStyle -> {FontFamily -> "Times"}]  ]
   , {\[Tau], 0, 7.3, 0.1}];
ListAnimate[p1]