Domanda del libro di testo
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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TOC.tableOfContents
- 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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 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?
Guida verificata passo dopo passo1
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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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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Percorso guidato
08:17
Acceleration in 2D
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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Percorso guidato
06:18Intro to Angular Momentum
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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Percorso guidato
11:10Converting Between Linear & Rotational
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