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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

8. Centripetal Forces & Gravitation

Uniform Circular Motion

Physics

8. Centripetal Forces & Gravitation

Uniform Circular Motion

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1:59 minutes

Problem 100c

Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f. Can the bead ride as high as the center of the circle (θ = 90°)? Explain.

Verified step by step guidance

Understand the problem: The bead is constrained to move inside a rotating vertical hoop. The question asks whether the bead can reach the center of the circle (θ = 90°), where θ is the angle measured from the bottom of the hoop. At this position, the bead would be at the same height as the center of the hoop.

Analyze the forces acting on the bead: At any position, the forces acting on the bead are (1) the gravitational force \( F_g = m g \), acting downward, and (2) the normal force exerted by the hoop, which provides the centripetal force required for circular motion. Additionally, the bead experiences a centrifugal force \( F_c = m \omega^2 r \sin(\theta) \) in the rotating reference frame, where \( \omega = 2 \pi f \) is the angular velocity.

Set up the condition for equilibrium at θ = 90°: For the bead to remain at θ = 90°, the net force in the radial direction must provide the centripetal force. At this position, the gravitational force \( m g \) acts perpendicular to the hoop, and the centrifugal force \( F_c \) acts outward along the radius. The normal force must balance these forces to keep the bead in equilibrium.

Write the force balance equation: At θ = 90°, the centrifugal force is \( F_c = m \omega^2 r \), and the gravitational force is \( F_g = m g \). For the bead to stay at this position, the normal force must satisfy \( N = F_c - F_g \). If \( F_c \) is less than \( F_g \), the bead cannot reach this position because the normal force would become negative, which is unphysical.

Determine the condition for the bead to reach θ = 90°: The centrifugal force must be at least equal to the gravitational force for the bead to reach this height. This gives the condition \( m \omega^2 r \geq m g \), or equivalently \( \omega^2 \geq \frac{g}{r} \). Substituting \( \omega = 2 \pi f \), the condition becomes \( (2 \pi f)^2 \geq \frac{g}{r} \), or \( f \geq \frac{1}{2 \pi} \sqrt{\frac{g}{r}} \). If this condition is met, the bead can reach θ = 90°.

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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 required to keep an object moving in a circular path, directed towards the center of the circle. In the context of the bead in the hoop, this force is necessary to counteract the gravitational force acting on the bead and maintain its circular motion. The balance between gravitational force and the required centripetal force determines the maximum height the bead can achieve within the hoop.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically calculated as U = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a reference point. As the bead rises within the hoop, its potential energy increases, which must be balanced by the kinetic energy and the forces acting on it to determine if it can reach the center of the hoop.

Equilibrium of Forces

Equilibrium of forces occurs when the net force acting on an object is zero, resulting in no acceleration. For the bead to ride at the center of the hoop (θ = 90°), the forces acting on it, including gravitational force and the normal force from the hoop, must balance perfectly. Analyzing these forces helps determine whether the bead can maintain this position without sliding down due to gravity.

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Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5–50). If his arms are capable of exerting a force of 1350 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 kg and the vine is 4.8 m long.

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Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f. Determine the angle θ where the bead will be in equilibrium within the hoop—that is, where it will have no tendency to move up or down along the hoop.

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Textbook Question

A train traveling at a constant speed rounds a curve of radius 215 m. A lamp suspended from the ceiling swings out to an angle of 18.5° throughout the curve. What is the speed of the train? [Hint: See Example 4–15.]

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Textbook Question

A small bead of mass m is constrained to slide without friction inside a circular vertical hoop of radius r which rotates about a vertical axis (Fig. 5–58) at a frequency f. If ƒ = 2.00 rev/s and r = 25.0cm, what is θ?

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Textbook Question

In a “Rotor-ride” at a carnival, people rotate in a vertical cylindrically walled “room.” See Fig. 5–52. If the room radius is 5.5 m, and the rotation frequency 0.50 revolutions per second when the floor drops out, what minimum coefficient of static friction keeps the people from slipping down? People on this ride said they were “pressed against the wall.” Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides nausea)? [Hint: Draw a free-body diagram for a person.]

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A flat puck (mass M) is revolved in a circle on a frictionless air hockey table top, and is held in this orbit by a light cord which is connected to a dangling mass (mass m) through a central hole as shown in Fig. 5–51. Show that the speed of the puck is given by v = √(mgR/M).

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Multiple Choice

What is the primary difference between motion along a circular pathway and motion along a linear pathway?

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Multiple Choice

In the context of uniform circular motion, which term describes a path that is not centered around a common point?

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