Rescale root of quartic equation

I have a quartic equation $1-r+frac{1}{w^2}r^4=0$ where I am interested in the plot of $frac{1}{w} f(r_0)$ (equal to $f(r_0/w)$) where $r_0$ is one of the roots. I would like to rescale the roots of the equations by $w^{-1}$ so that I don’t have this factor of $frac{1}{w}$ in my equation. However when I do $rto wtilde{r}$ I do not get the same plot anymore. What am I doing wrong?

S = Solve[1 - r + 1/w^2*r^4 == 0, r];
Plot[Abs[1/w*Im[S[[3, 1, 2]]]], {w, -wc, wc}]

S2 = Solve[1 - w*r + w^2*r^4 == 0, r];

Plot[Abs[Im[S2[[3, 1, 2]]]], {w, -wc, wc}]

Note that when I apply this procedure with a cubic $1-r+frac{1}{w^2}r^3=0$ then the above works perfectly fine.