 Solution

For horizontal forcing the exact angular equation isWrite $\theta =\Theta +\xi$ with $\Theta ={\theta }_1$ slowly varying and $\xi =-(a/l)\cos \Theta \cos \omega t$. To leading order in the slow-to-fast frequency ratio, $\ddot{\xi }=(a{\omega }^2/l)\cos \Theta \cos \omega t$, which balances the rapid forcing. Expand the right-hand side to first order in $\xi$:During one fast period treat $\Theta$ as fixed. The averages $⟨\xi ⟩$ and $⟨\cos \omega t⟩$ vanish, whereasThe leading averaged equation is thereforeThe effective force comes from correlation between the rapid displacement and the rapidly varying driving torque; simply averaging the driving cosine to zero misses it. This keeps the leading nonzero fast-motion contribution, proportional to $a^2{\omega }^2$, while neglecting terms suppressed by additional powers of $a/l$ or of the slow rate divided by $\omega$. The gravitational expansion at order $g(a/l{)}^2$ is smaller in the stated high-frequency regime. The resulting slow frequencies must also remain small compared with $\omega$ for averaging to be self-consistent.