A car drives with constant speed on an elliptical track, as...

A car drives with constant speed on an elliptical track, as shown in Figure 9.29. Rank the points A, B, and $C$ in order of increasing likelihood that the car might skid. Indicate ties where appropriate.

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Key Concepts

Friction and Skidding

Friction between the tires and the track provides the necessary force to supply the centripetal acceleration. If the frictional force available is less than the required centripetal force, the car will skid. This interplay between friction and the demands of centripetal force is central to understanding where skidding is likely to occur.

Centripetal Acceleration

This concept refers to the acceleration directed toward the center of a curved path that is necessary to keep an object moving along that path. It is determined by the formula v²/r, meaning that at a constant speed, a tighter curve (smaller r) requires a greater centripetal force, which is directly linked to the risk of skidding.

Radius of Curvature

The radius of curvature indicates how sharply a path turns at any point. In the context of an elliptical track, the radius varies from point to point; a smaller radius signifies a sharper bend and thus a larger required centripetal force. This makes sections with smaller radii more prone to skidding.

Elliptical Track Geometry

An ellipse has non-uniform curvature; it typically curves more sharply at the ends of the minor axis compared to the areas near the major axis. Recognizing this geometric property is crucial for analyzing how the risk of skidding changes along different sections of an elliptical track.

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