A roller-coaster car at an amusement park has a mass of 1200 kg when fully loaded with passenger

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A roller-coaster car at an amusement park has a mass of 1200...

A roller-coaster car at an amusement park has a mass of 1200 $\mathrm{kg}$ when fully loaded with passengers. As the car passes over the top of a circular hill of radius $18 \mathrm{m},$ assume that its speed is not changing. At the top of the hill, what are the (a) magnitude $F_{N}$ and (b) direction (up or down) of the normal force on the car from the track if the car's speed is $v=11 \mathrm{m} / \mathrm{s} ?$ What are $(\mathrm{c}) F_{N}$ and (d) the direction if $v=14 \mathrm{m} / \mathrm{s} ?$

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Key Concepts

Centripetal Acceleration

This is the acceleration that points toward the center of the circle in circular motion. Its magnitude is given by the formula v²/r, where v is the speed of the object and r is the radius of the circular path. It is essential in analyzing how forces must be arranged to sustain circular motion, ensuring that the object remains on its curved path.

Normal Force

The normal force is the force exerted by a surface perpendicular to itself, counteracting other forces such as gravity. In scenarios involving circular motion, such as at the top of a hill on a roller coaster, the normal force combines with gravitational force to provide the necessary centripetal force for the circular path. Its magnitude and direction may change depending on the speed and curvature of the track.

Newton's Second Law in Circular Motion

Newton's second law states that the net force on an object equals its mass multiplied by its acceleration. When applied to objects in circular motion, this law requires that the difference between gravitational force and the normal force provides the needed centripetal force (m(v²/r)) directed towards the center of the circle. This is fundamental in problems where the forces at a specific point on a curved path, like at the top of a hill, need to be determined.

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