A passenger with mass 85 kg rides in a Ferris wheel like that in Example 5.24( Section 5.4) . T

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A passenger with mass 85 kg rides in a Ferris wheel like...

A passenger with mass 85 $\mathrm{kg}$ rides in a Ferris wheel like that in Example 5.24$(\text { Section } 5.4)$ . The seats travel in a circle of radius 35 $\mathrm{m}$ . The Ferris wheel rotates at constant speed and makes one complete revolution every 25 $\mathrm{s}$ . Calculate the magnitude and direction of the net force exerted on the passenger by the seat when she is (a) one-quarter revolution past her lowest point and (b) one-quarter revolution past her highest point.

Key Concepts

Vector Decomposition of Forces

Vector decomposition involves breaking down forces into their constituent components, typically in orthogonal directions such as horizontal and vertical or radial and tangential. This technique is crucial for correctly calculating the net force on an object when multiple forces act in different directions and helps in understanding how each component contributes to the overall motion.

Gravitational Force

Gravitational force is the attractive force exerted by the Earth, which acts downward on any mass with a magnitude of mg, where m is the mass of the object and g is the acceleration due to gravity. This force must be considered along with other forces, such as the centripetal force, especially when the object is in motion in a gravitational field.

Free-Body Diagrams

A free-body diagram is a visual tool used to represent all the forces acting on an object. By isolating the object and sketching vectors representing forces like gravitational force and the force providing centripetal acceleration, one can systematically analyze the net force and its direction. This diagram is fundamental in solving problems that involve the interaction of several forces, especially in settings like uniform circular motion.

Uniform Circular Motion

Uniform circular motion occurs when an object travels along a circular path at a constant speed. Although the speed remains constant, the object is accelerating because its direction is continuously changing. This acceleration is always directed toward the center of the circle, and understanding this concept is key when analyzing forces in such systems.

Centripetal Force and Acceleration

The centripetal force is the net force required to keep an object moving in a circle. It is calculated by the formula a = v²/r (or in terms of period, a = 4?²r/T²), where v is the speed, r is the radius, and T is the period of revolution. This force always points toward the center of the circle, and it is essential to consider it when analyzing forces acting on an object in circular motion.

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