f[t_] = (Sin[2.5*(t+k)])^3 + (Cos[2.5*(t+k)])^3
tt = Table[k = 7/75*n;
PolarPlot[f[t],{t,0,14},PlotRange->{{-1.1,1.1},{-1.1,1.1}},PlotPoints-> 500,PlotStyle-> RGBColor[1,0.65,0],Background->RGBColor[1,1,0.94],ImageSize-> 350, AspectRatio-> 1]
,{n,0,60}];
ListAnimate[tt]
Export["gif.gif",tt,"AnimationRepetitions"->Infinity]

g[t_] = Sin[k*t/5]
TT= Table[k=n/150;
PolarPlot[g[t], {t, 0, 56},PlotPoints->500,PlotStyle-> RGBColor[1,0.65,0],Background->RGBColor[1,1,0.94],ImageSize->350,AspectRatio->1]
,{n,1,300}];
ListAnimate[TT]