If the 2 -kg block passes over the top B of the circular...

If the 2 -kg block passes over the top $B$ of the circular portion of the path with a speed of $3.5 \mathrm{m} / \mathrm{s}$ calculate the magnitude $N_{B}$ of the normal force exerted by the path on the block. Determine the maximum speed $v$ which the block can have at $A$ without losing contact with the path.

Key Concepts

Force Analysis in Circular Motion

Performing a force analysis, often through free-body diagrams, is a key method in solving circular motion problems. By identifying and resolving forces such as gravity and the normal force, one can set up equations that equate the sum of forces toward the center of the circle with the mass times the required centripetal acceleration. This systematic approach is essential for determining unknown quantities such as the magnitude of forces or speeds at specific points along the path.

Critical Speed and Loss of Contact

Critical speed refers to the maximum speed an object can have at a point along a curved path without losing contact with the surface. This condition occurs when the normal force becomes zero, meaning that gravity alone is sufficient to provide the necessary centripetal force. Understanding this concept is vital in assessing safety and designing systems like roller coasters where maintaining contact is crucial.

Centripetal Acceleration

Any object moving in a circular path experiences an acceleration directed toward the center of the circle, known as centripetal acceleration. This acceleration is essential to keep the object moving along the curved path and is calculated by the square of the object's speed divided by the radius of the circle. It underpins the analysis of forces on an object in circular motion, as the net inward force must equal the mass times the centripetal acceleration.

Normal Force in Circular Motion

The normal force is the perpendicular reactive force that a surface exerts on an object in contact with it. In the context of circular motion, especially at the top of a loop, the normal force works in combination with the gravitational force to provide the required net inward force. The magnitude of this force adjusts based on the object’s speed and curvature of the path, and its reduction to zero indicates the onset of the object losing contact with the surface.

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