Refer back to Fig. 9-6. How large must h be (in terms of R )...

Refer back to Fig. 9-6. How large must $h$ be (in terms of $R$ ) if the frictionless wire is to exert no force on the bead as it passes through point- $B$ ? Assume the bead is released from rest at $A$.

Key Concepts

Centripetal Force Requirement

An object moving along a curved path requires a centripetal force to continually change direction. In frictionless scenarios, the needed centripetal force must be provided by other forces such as components of gravity or the normal force. Analyzing this requirement helps in understanding conditions like free flight or loss of contact along a circular path.

Conditions for Loss of Contact

Loss of contact occurs when the supporting force (normally the normal force) becomes zero, meaning the object no longer follows the curvature of the track. In such cases, the remaining force components, such as gravity, must provide the precise centripetal force needed for the motion. Establishing this condition is key to solving problems where the constraint force vanishes at a specific point.

Conservation of Energy

This principle states that in a frictionless system, the total energy (potential plus kinetic) remains constant. In problems involving sliding along curves, it allows one to equate the potential energy loss of a bead to its gain in kinetic energy, thereby providing a way to determine the speed of the bead at various positions along its path.

Normal Force and Constrained Motion

When an object moves along a prescribed path, such as a bead on a wire, the normal force is the force exerted by the surface to keep the object on its track. It is a crucial factor in constrained motion, and determining conditions when this force is zero reveals when the object is no longer in contact with the surface.

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