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Ch 08: Dynamics II: Motion in a Plane
Chapter 8, Problem 8

In an amusement park ride called The Roundup, passengers stand inside a 16-m-diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in FIGURE P8.51. b. What is the longest rotation period of the wheel that will prevent the riders from falling off at the top?

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. In the context of the amusement park ride, this force is necessary to keep the riders moving in a circular motion. It is provided by the gravitational force acting on the riders and the normal force from the ride's structure.
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Gravitational Force

Gravitational force is the attractive force between two masses, which in this case is the force acting on the riders due to Earth's gravity. At the top of the ride, this force must be sufficient to provide the necessary centripetal force to keep the riders from falling off. The gravitational force can be calculated using the equation F = mg, where m is the mass and g is the acceleration due to gravity.
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Period of Rotation

The period of rotation is the time it takes for one complete revolution around a circular path. It is related to the angular velocity and the radius of the circular motion. In this scenario, the longest rotation period corresponds to the slowest speed at which the ride can operate while still providing enough centripetal force to keep the riders safely in place at the top of the loop.
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Related Practice
Textbook Question
A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) (a) What is the gravitational force acting on the ball?
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Textbook Question
A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) (b) What is the tension in the string when the ball is at the top?
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Textbook Question
A heavy ball with a weight of 100 N (m = 10.2 kg) is hung from the ceiling of a lecture hall on a 4.5-m-long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.5 m/s as it passes through the lowest point. What is the tension in the rope at that point?
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Textbook Question
Suppose you swing a ball of mass m in a vertical circle on a string of length L. As you probably know from experience, there is a minimum angular velocity ωₘᵢₙ you must maintain if you want the ball to complete the full circle without the string going slack at the top. a. Find an expression for ωₘᵢₙ.
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Textbook Question
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A 1500 kg car takes a 50-m-radius unbanked curve at 15 m/s. What is the size of the friction force on the car?
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