Figure (8-E14) shows a light rod of length l rigidly...

Figure (8-E14) shows a light rod of length $l$ rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth $h$ below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of $h$ so that the block moves in a complete circle about the ring?

Key Concepts

Critical Conditions for Complete Circular Motion

Completing a full rotation in vertical circular motion imposes minimum speed conditions at critical points, particularly at the top of the circle. At the highest point, the net required inward force must be sufficiently provided by the gravitational force and any other physical constraints (such as tension in a rod). Formulating the energy and force conditions that meet these minimum requirements is essential to ensure continuous contact and to complete the circular path.

Conservation of Mechanical Energy

This principle indicates that in the absence of non-conservative forces (such as friction), the sum of kinetic and potential energy in a system remains constant. In dynamics problems, it allows us to relate changes in height (and thus gravitational potential energy) to changes in speed (kinetic energy), enabling predictions of motion parameters at different positions in a path.

Centripetal Force Requirements

For an object to follow a circular path, a net inward (centripetal) force is essential. This force, expressed as mv²/r, ensures that the object remains on its curved trajectory. Understanding the balance between this requirement and the forces available (like gravity or tension) is critical in analyzing whether an object can complete a loop without losing contact or slackening the connection.

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