A boy is initially seated on the top of a hemispherical ice...

A boy is initially seated on the top of a hemispherical ice mound of radius $R=13.8 \mathrm{~m} . \mathrm{He}$ begins to slide down the ice, with a negligible initial speed (Fig. 8-47). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

Key Concepts

Loss of Contact Condition

The loss of contact occurs when the normal force becomes zero, meaning the surface ceases to provide the necessary centripetal force to keep the object in circular motion. At this point, the only force acting on the object is gravity, and from the balance of forces, one can determine the specific location along the curve where this transition occurs.

Normal Force

The normal force is the reactive force exerted by a surface perpendicular to the direction of contact. For an object sliding on a curved frictionless surface, it is the force that maintains contact with the surface. When the normal force diminishes to zero, it indicates that the object can no longer maintain its circular trajectory.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. In the context of an object sliding on a curved surface, this force is provided by the component of gravitational force and the normal force. Analyzing how these forces balance provides insight into the object's motion along the curve.

Conservation of Energy

In frictionless motion, energy is conserved as the gravitational potential energy of the system converts into kinetic energy as the object descends. This principle allows us to equate the loss in potential energy to the gain in kinetic energy and is essential for analyzing the speed of the object at various points along its path.

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