A roller-coaster car has a mass of 1200 kg when fully loaded with passengers. As the car passes

step 1 So basically we have given roller coaster car having a mass of 1200 kg and it is moving on the s

A roller-coaster car has a mass of 1200 kg when fully...

A roller-coaster car 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 $18 \mathrm{~m}$, its speed is not changing. What are the magnitude and direction of the force of the track on the car at the top of the hill if the car's speed is (a) $11 \mathrm{~m} / \mathrm{s}$ and (b) $14 \mathrm{~m} / \mathrm{s}$ ?

Key Concepts

Newton's Second Law

This principle states that the net force acting on an object is equal to its mass multiplied by its acceleration. When applied to circular motion, it relates all the forces acting on an object (such as gravitational force and the normal force) to the centripetal acceleration required to maintain its curved path.

Centripetal Acceleration

This is the acceleration directed toward the center of a circle that an object needs to maintain circular motion. It is calculated using the formula a = v²/r, where v is the object’s speed and r is the radius of the circle, and it must be provided by the net force acting on the object.

Normal Force

This is the contact force exerted by a surface, acting perpendicular to that surface. In the context of circular motion over a hill, the normal force from the track combines with gravitational force to yield the net force necessary for centripetal acceleration, and its magnitude and direction may change depending on the speed of the object.

Gravitational Force

This is the force with which the Earth attracts an object, calculated as the mass of the object multiplied by the acceleration due to gravity. It acts downward and plays a crucial role in circular motion problems by contributing to the net force needed for centripetal acceleration, especially when the object is moving over curved surfaces like hills.

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