What does this mean when E=U_eff? I don't think this means the first term in E is zero. I don't understand the sentence ' This is a cubic equation for cos(theta)'
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It means there is an asymptote for $$ E=U_{eff}(\theta) $$ which corresponds to a certain value of $\theta$ you can evaluate by solving the equation: $$ E=\frac{M_z^2}{2ml^2}\sin^2{\theta}-mgl\cos{\theta} $$ So $E$ takes this value in a infinite time. This equation is quadratic in $\cos{\theta}$ also because it's written that there are 2 roots. Maybe it's a misprint.
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$\begingroup$ Hi @Giuseppe Galesi, just a note: the time needed to obtain $U_{\text{eff}} (\theta ) = E$ needs not to be infinite - indeed, it's infinite if and only if the inversion point is an equilibrium point ($\frac {dU}{d\theta}=0$). $\endgroup$– pppqqqCommented May 1, 2015 at 8:39