🎬Centrifugal Force as the Stability Governor

The analysis across the three scenarios demonstrates that the system's dynamic fate is entirely determined by the initial balance of forces. The system achieves a near-perfect circular orbit (Scenario C) only when the initial centrifugal force exactly counteracts the gravitational tension from the hanging mass $m_2$. If the initial angular velocity is too low (Scenario A), the centrifugal force is insufficient, and $m_2$ 's weight pulls $m_1$ rapidly inward toward collapse due to the dominant inward force. Conversely, if the initial angular velocity is too high (Scenario B), the centrifugal force is overpowering, flinging $m_1$ outward and initiating a continuous, large amplitude coupled oscillation as the forces repeatedly overshoot equilibrium.

Finding the Generalized Inertia Tensor for the Coupled Mass System | Proof and Derivationviadean on Notion