Textbook Question
An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What fraction of its kinetic energy is rotational?
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Ch 12: Rotation of a Rigid Body
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 12, Problem 33b
A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. What is the speed of a point at the top edge of the tire?
Verified step by step guidance1
Step 1: Understand the problem. The car tire is rotating while the car is moving forward. The speed of a point at the top edge of the tire is the sum of the linear speed of the car and the tangential speed of the tire due to its rotation.
Step 2: Calculate the radius of the tire. The diameter is given as 60 cm, so the radius \( r \) is half of the diameter: \( r = \frac{60}{2} = 30 \, \text{cm} = 0.3 \, \text{m} \).
Step 3: Determine the angular velocity \( \omega \) of the tire. The linear speed of the car \( v \) is related to the angular velocity by the formula \( v = r \omega \). Rearrange to find \( \omega = \frac{v}{r} \). Substitute \( v = 20 \; \text{m/s} \) and \( r = 0.3 \; \text{m} \).
Step 4: Calculate the tangential speed of a point on the tire due to its rotation. The tangential speed at the top edge of the tire is equal to \( r \omega \), which is the same as the linear speed of the car, \( v = 20 \; \text{m/s} \).
Step 5: Add the linear speed of the car and the tangential speed of the tire at the top edge. The total speed of a point at the top edge of the tire is \( v_{\text{top}} = v + v = 20 + 20 = 40 \; \text{m/s} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Speed
Linear speed refers to the distance traveled per unit of time. In this context, it is the speed at which the car is moving, which is given as 20 m/s. This speed is crucial for understanding how the motion of the car translates to the motion of points on the tire.
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Rotational Motion
Rotational motion describes how an object rotates around an axis. For a tire, this involves the relationship between its linear speed and the angular speed, which is determined by the radius of the tire. Understanding this concept helps in calculating the speed of points on the tire as it rolls.
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Tangential Speed
Tangential speed is the linear speed of a point on the circumference of a rotating object. At the top edge of the tire, the tangential speed is the sum of the car's linear speed and the speed due to the tire's rotation. This concept is essential for determining the speed of a point at the top edge of the tire.
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Related Practice
Textbook Question
The object shown in FIGURE EX12.29 is in equilibrium. What are the magnitudes of and ?
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Textbook Question
A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. What is the speed of a point at the bottom edge of the tire?
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Textbook Question
A 5.0 kg cat and a 2.0 kg bowl of tuna fish are at opposite ends of the 4.0-m-long seesaw of FIGURE EX12.32. How far to the left of the pivot must a 4.0 kg cat stand to keep the seesaw balanced?
Textbook Question
An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What is the sphere’s angular velocity at the bottom of the incline?
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Textbook Question
A 12-cm-diameter, 600 g cylinder, initially at rest, rotates on an axle along its axis. A steady 0.50 N force applied tangent to the edge of the cylinder causes the cylinder to reach an angular velocity of 500 rpm in 2.0 s. What is the magnitude of the frictional torque between the cylinder and the axle?
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