Textbook Question
Starting from rest, a DVD steadily accelerates to 500 rpm in 1.0 s, rotates at this angular speed for 3.0 s, then steadily decelerates to a halt in 2.0 s. How many revolutions does it make?
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Ch 04: Kinematics in Two Dimensions
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 4, Problem 35
Your roommate is working on his bicycle and has the bike upside down. He spins the 60-cm-diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. What are the pebble's speed and acceleration?
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Determine the radius of the wheel. The diameter is given as 60 cm, so the radius \( r \) is half of the diameter: \( r = \frac{60}{2} = 30 \, \text{cm} = 0.30 \, \text{m} \).
Calculate the angular velocity \( \omega \) of the wheel. The pebble completes 3 revolutions per second, so the angular velocity is \( \omega = 2 \pi \times 3 = 6 \pi \, \text{rad/s} \).
Find the pebble's linear speed \( v \). The relationship between linear speed and angular velocity is \( v = r \omega \). Substitute \( r = 0.30 \, \text{m} \) and \( \omega = 6 \pi \): \( v = 0.30 \times 6 \pi \).
Determine the pebble's centripetal acceleration \( a_c \). The formula for centripetal acceleration is \( a_c = \frac{v^2}{r} \). Substitute \( v = 0.30 \times 6 \pi \) and \( r = 0.30 \): \( a_c = \frac{(0.30 \times 6 \pi)^2}{0.30} \).
Simplify the expressions for \( v \) and \( a_c \) to find the pebble's speed and acceleration. Ensure the units are consistent and the results are expressed in \( \text{m/s} \) for speed and \( \text{m/s}^2 \) for acceleration.
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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 by an object per unit of time. In this scenario, the pebble's speed can be calculated by determining the circumference of the bicycle wheel and multiplying it by the number of revolutions per second. The formula for linear speed (v) is v = distance/time, where the distance is the circumference of the wheel.
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Centripetal Acceleration
Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It can be calculated using the formula a_c = v²/r, where v is the linear speed and r is the radius of the circular path. This concept is crucial for understanding how the pebble accelerates as it moves along the wheel's circular trajectory.
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Angular Velocity
Angular velocity is a measure of how quickly an object rotates around a central point, expressed in radians per second. It is related to the number of revolutions per second and can be calculated using the formula ω = 2πf, where f is the frequency of rotation. In this case, knowing the angular velocity helps in determining both the linear speed and the centripetal acceleration of the pebble.
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Related Practice
Textbook Question
FIGURE EX4.36 shows the angular velocity graph of the crankshaft in a car. What is the crankshaft's angular acceleration at t = 3s
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Textbook Question
The radius of the earth's very nearly circular orbit around the sun is 1.5 x 1011 m. Find the magnitude of the earth's velocity.
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Textbook Question
A speck of dust on a spinning DVD has a centripetal acceleration of 20 m/s3 . What would be the acceleration of the first speck of dust if the disk's angular velocity was doubled?
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Textbook Question
The radius of the earth's very nearly circular orbit around the sun is 1.5 x 1011 m. Find the magnitude of the earth's angular velocity.
Textbook Question
A 5.0-m-diameter merry-go-round is initially turning with a 4.0 s period. It slows down and stops in 20 s. How many revolutions does the merry-go-round make as it stops?