Textbook Question
A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its center with 0.15 J of kinetic energy. What is the speed of a point on the rim?
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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 4
An 18-cm-long bicycle crank arm, with a pedal at one end, is attached to a 20-cm-diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s. What is the tangential acceleration of a tooth on the sprocket?
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Step 1: Identify the given quantities and their units. The length of the crank arm is 18 cm, the diameter of the sprocket is 20 cm (radius = 10 cm), the initial pedaling rate is 60 rpm, the final pedaling rate is 90 rpm, and the time interval for the change is 10 s.
Step 2: Convert the pedaling rates from revolutions per minute (rpm) to angular velocity in radians per second. Use the formula \( \omega = \frac{2\pi \times \text{rpm}}{60} \). Calculate \( \omega_{\text{initial}} \) and \( \omega_{\text{final}} \).
Step 3: Determine the angular acceleration \( \alpha \) of the sprocket using the formula \( \alpha = \frac{\omega_{\text{final}} - \omega_{\text{initial}}}{\Delta t} \), where \( \Delta t \) is the time interval (10 s).
Step 4: Relate the tangential acceleration \( a_{\text{tangential}} \) to the angular acceleration \( \alpha \) using the formula \( a_{\text{tangential}} = \alpha \cdot r \), where \( r \) is the radius of the sprocket (10 cm).
Step 5: Substitute the values for \( \alpha \) and \( r \) into the formula for \( a_{\text{tangential}} \) to express the tangential acceleration of a tooth on the sprocket.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangential Acceleration
Tangential acceleration refers to the rate of change of tangential velocity of an object moving along a circular path. It is calculated as the product of the radius of the circular path and the angular acceleration. In this context, it helps determine how quickly a point on the sprocket is speeding up as the cyclist increases her pedaling rate.
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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 or revolutions per minute (rpm). In this scenario, the cyclist's pedaling rate is given in rpm, which must be converted to angular velocity to analyze the motion of the sprocket and its teeth effectively.
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Relationship Between Linear and Angular Quantities
The relationship between linear and angular quantities is fundamental in rotational motion. The linear velocity (v) of a point on a rotating object is related to the angular velocity (ω) and the radius (r) by the equation v = rω. This relationship is crucial for calculating the tangential acceleration of the sprocket's teeth as the cyclist changes her pedaling rate.
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Related Practice
Textbook Question
A high-speed drill reaches 2000 rpm in 0.50 s. What is the magnitude of the drill's angular acceleration?
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A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. What is the speed of the tip of a blade 10 s after the fan is turned off?
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Textbook Question
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