rot[phi_] := {{Cos[phi], -Sin[phi]}, {Sin[phi], Cos[phi]}};
omegaPendulum = 1.0;
omegaEarth = 0.2;
xy[t_] := {Cos[omegaPendulum*t], 0};
xyRot[t_] := rot[omegaEarth*t] . xy[t];
ParametricPlot[xyRot[t], {t, 0, 10 Pi}]

(* This is more relevant to HW problem 5 *)
omega = 0.2;
x0 = 0;
v = 1;
tmax = 30;
ParametricPlot[rot[omega*t] . {x0+ v*t, 0}, {t, 0, tmax}]

ClearAll["Global`*"];
g = 9.8; r = 5.0; omegasq = Sqrt[g/r]; phi0 = 20*Degree;
soln = NDSolveValue[
{phi''[t] == -omegasq*Sin[phi[t]], 
phi'[0] == 0, phi[0] == phi0}, phi[t], {t, 0, 10}];
Show[
{Plot[soln, {t, 0, 10}],
Plot[phi0*Cos[Sqrt[omegasq]*t], {t, 0, 10}]}]