The fast French train known as the TGV (Train à Grande...

The fast French train known as the TGV (Train à Grande Vitesse) has a scheduled average speed of $216 \mathrm{~km} / \mathrm{h} .$ (a) If the train goes around a curve at that speed and the magnitude of the acceleration experienced by the passengers is to be limited to $0.050 \mathrm{~g}$, what is the smallest radius of curvature for the track that can be tolerated? (b) At what speed must the train go around a curve with a $1.00 \mathrm{~km}$ radius to be at the acceleration limit?

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration required to maintain an object in circular motion. It is given by the formula a = v²/r, where v is the speed and r is the radius of the circular path. This concept is key in determining the forces acting on objects as they follow curved trajectories.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or curve. In this type of motion, although the speed may remain constant, the direction changes continuously, resulting in an acceleration directed toward the center of the circle, known as centripetal acceleration.

Unit Conversion

Unit conversion is essential when working with different measurement systems. In problems involving speeds and accelerations, it is often necessary to convert speeds from kilometers per hour to meters per second and express accelerative forces in standard units such as meters per second squared, especially when comparing them to the acceleration due to gravity.

Relationship Between Speed, Radius, and Acceleration

The formula a = v²/r encapsulates the relationship between an object's speed, the radius of the curvature, and the resulting centripetal acceleration. This relationship is crucial when determining limits such as maximum allowable speeds or minimum curve radii to ensure that acceleration remains within safe bounds.

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