Book cover for College Physics Explore and Apply

College Physics Explore and Apply

Eugenia Etkina; Alan Van Heuvelen; Gorazd Planinši?

ISBN #9780134601823

2nd Edition

2,244 Questions

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46,660 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Circular motion involves objects moving along curved paths with constant speed, yet undergoing acceleration due to continuous changes in the direction of their velocity. The forces acting on such objects always have a net inward (centripetal) component, as captured in the expression ar = v²/r and its period-based form ar = 4?²r/T². These principles extend to gravitational interactions governing planetary orbits and satellite motion, illustrating a unifying theme in dynamics from everyday experiences to celestial mechanics.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Atomic Physics

The study of atomic models and structure, detailing the evolution from classical orbit models (e.g., Bohr’s model) to quantum mechanics with quantized energy levels, quantum numbers, the uncertainty principle, and tunneling.

Example Problems

Example 1

While mountain biking, you first move at constant speed along the bottom of a trail 's circular dip and then at constant speed across the top of a circular hump. Assume that you and the bike are a system. Determine the direction of the acceleration at each position and construct a force diagram for each position (consistent with the direction of the acceleration). Compare at each position the magnitude of the force of the surface on the bike with the force Earth exerts on the system.

Example 2

You swing a rock tied to a string in a vertical circle. (a) Determine the direction of the acceleration of the rock as it passes the lowest point in its swing. Construct a consistent force diagram for the rock as it passes that point. How does the force that the string exerts on the rock compare to the force that Earth exerts on the rock? Explain. (b) Repeat this analysis as best you can for the rock as it passes the highest point in the swing. (c) If the string is tied around your finger, when do you feel a stronger pull-when the rock is at the bottom of the swing or at the top? Explain.

Example 3

You ride a roller coaster with a loop-the-loop. Compare as best you can the normal force that the seat exerts on you to the force that Earth exerts on you when you are passing the bottom of the loop and the top of the loop. Justify your answers by determining the direction of acceleration and constructing a force diagram for each position. Make your answers consistent with Newton's second law.

Example 4

You start an old record player and notice a bug on the surface close to the edge of the record. The record has a diameter of 12 inches and completes 33 revolutions each minute. (a) What are the speed and the acceleration of the bug? (b) What would the bug"s speed and acceleration be if it were halfway between the center and the edge of the record?

Example 5

Determine the acceleration of Earth due to its motion around the Sun. What do you need to assume about Earth to make the calculation? How does this acceleration compare to the acceleration of free fall on Earth?
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Step-by-Step Explanations

QUESTION

How can you determine the direction and magnitude of the acceleration of an object moving in a circle at constant speed?\nStep-by-step Answer:\nStep 1: Identify the point of interest on the circular path and draw the instantaneous velocity vector (tangent to the circle) just before the point.\nStep 2: Draw the instantaneous velocity vector just after the point, ensuring both velocity vectors are drawn with their tails at the same point.\nStep 3: Draw the velocity change vector (\u0394v) from the head of the initial velocity to the head of the final velocity.\nStep 4: Determine the acceleration vector by dividing \u0394v by the time interval (\u0394t). The acceleration will point in the same direction as \u0394v.\nStep 5: Recognize that for constant speed circular motion, the acceleration must point toward the center of the circle, confirming the concept of centripetal (radial) acceleration.\n\n- Topic: Deriving the Relationship Between Speed, Radius, and Period \nQuestion: How do you derive the expression for radial acceleration in terms of the period T of circular motion?\nStep-by-step Answer:\nStep 1: Start with the definition of speed in circular motion: v = 2\u03c0r/T, where r is the radius and T is the period.\nStep 2: Substitute the expression for v into the centripetal acceleration formula: ar = v\u00b2/r.\nStep 3: Replace v with (2\u03c0r/T) to obtain ar = (2\u03c0r/T)\u00b2/r.\nStep 4: Simplify the expression: ar = 4\u03c0\u00b2r/T\u00b2.\nStep 5: Conclude that the radial acceleration is directly proportional to the radius and inversely proportional to the square of the period.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Identify the point of interest on the circular path and draw the instantaneous velocity vector (tangent to the circle) just before the point.\nStep 2: Draw the instantaneous velocity vector just after the point, ensuring both velocity vectors are drawn with their tails at the same point.\nStep 3: Draw the velocity change vector (\u0394v) from the head of the initial velocity to the head of the final velocity.\nStep 4: Determine the acceleration vector by dividing \u0394v by the time interval (\u0394t). The acceleration will point in the same direction as \u0394v.\nStep 5: Recognize that for constant speed circular motion, the acceleration must point toward the center of the circle, confirming the concept of centripetal (radial) acceleration.\n\n- Topic: Deriving the Relationship Between Speed, Radius, and Period \nQuestion: How do you derive the expression for radial acceleration in terms of the period T of circular motion?\nStep-by-step Answer:\nStep 1: Start with the definition of speed in circular motion: v = 2\u03c0r/T, where r is the radius and T is the period.\nStep 2: Substitute the expression for v into the centripetal acceleration formula: ar = v\u00b2/r.\nStep 3: Replace v with (2\u03c0r/T) to obtain ar = (2\u03c0r/T)\u00b2/r.\nStep 4: Simplify the expression: ar = 4\u03c0\u00b2r/T\u00b2.\nStep 5: Conclude that the radial acceleration is directly proportional to the radius and inversely proportional to the square of the period.\n\n"
Final Answer:

"- Topic: Determining Radial Acceleration Using the Velocity Change Method \nQuestion: How can you determine the direction and magnitude of the acceleration of an object moving in a circle at constant speed?\nStep-by-step Answer:\nStep 1: Identify the point of interest on the circular path and draw the instantaneous velocity vector (tangent to the circle) just before the point.\nStep 2: Draw the instantaneous velocity vector just after the point, ensuring both velocity vectors are drawn with their tails at the same point.\nStep 3: Draw the velocity change vector (\u0394v) from the head of the initial velocity to the head of the final velocity.\nStep 4: Determine the acceleration vector by dividing \u0394v by the time interval (\u0394t). The acceleration will point in the same direction as \u0394v.\nStep 5: Recognize that for constant speed circular motion, the acceleration must point toward the center of the circle, confirming the concept of centripetal (radial) acceleration.\n\n- Topic: Deriving the Relationship Between Speed, Radius, and Period \nQuestion: How do you derive the expression for radial acceleration in terms of the period T of circular motion?\nStep-by-step Answer:\nStep 1: Start with the definition of speed in circular motion: v = 2\u03c0r/T, where r is the radius and T is the period.\nStep 2: Substitute the expression for v into the centripetal acceleration formula: ar = v\u00b2/r.\nStep 3: Replace v with (2\u03c0r/T) to obtain ar = (2\u03c0r/T)\u00b2/r.\nStep 4: Simplify the expression: ar = 4\u03c0\u00b2r/T\u00b2.\nStep 5: Conclude that the radial acceleration is directly proportional to the radius and inversely proportional to the square of the period.\n\n"

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Common Mistakes

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