FrontierScience / 70929247-a356-4a12-8534-49fd885f0c3e / Consider a pendulum with a length `l` and a mass `m` at the end, where the mass of the rod is negligible and the pendulum bob is treated as a point mass. The pendulum is placed inv…

Consider a pendulum with a length `l` and a mass `m` at the end, where the mass of the rod is negligible and the pendulum bob is treated as a point mass. The pendulum is placed inverted, with the bob at the top, and the lower end of the rod oscillates vertically with `y(t)=acosΩt`. Assume `\( a≪l \)` and ` Ω≫√(g/l)`, where `g` is the gravitational acceleration. Determine the condition for stable equilibrium of the angle between the rod and the vertical at `θ=0`. Your answer must in the form of an inequality for \\( \\Omega \\) in terms of \\( g \\), \\( l \\), and \\( a \\).

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

Consider a pendulum with a length `\( l \)` and a mass `\( m \)` at the end, where the mass of the rod is negligible and the pendulum bob is treated as a point mass. The pendulum is placed inverted, with the bob at the top, and the lower end of the rod oscillates vertically with `\( y(t)=a\cos\Omega t \)`. Assume `\( a≪l \)` and ` \( \Omega \gg \sqrt{g/l} \)`, where `\( g \)` is the gravitational acceleration. Determine the condition for stable equilibrium of the angle between the rod and the vertical at `\( θ=0 \)`. Your answer must in the form of an inequality for \\( \\Omega \\) in terms of \\( g \\), \\( l \\), and \\( a \\).

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.