Textbook Question
FIGURE EX4.24 shows the angular-position-versus-time graph for a particle moving in a circle. What is the particle's angular velocity at t = 4s
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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 28a
As the earth rotates, what is the speed of a physics student in Miami, Florida, at latitude 26°. Ignore the revolution of the earth around the sun. The radius of the earth is 6400 km.
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Determine the radius of the Earth at the equator, which is approximately 6378 km. Convert this value to meters for consistency in SI units: \( R_{\text{Earth}} = 6.378 \times 10^6 \ \text{m} \).
Calculate the effective radius of the circular path of the student in Miami. Since the student is at a latitude of 26 degrees, the effective radius is given by \( R_{\text{effective}} = R_{\text{Earth}} \cdot \cos(\theta) \), where \( \theta = 26^{\circ} \).
Determine the angular velocity of the Earth's rotation. The Earth completes one full rotation in 24 hours, so the angular velocity is \( \omega = \frac{2\pi}{T} \), where \( T = 24 \ \text{hours} = 24 \cdot 3600 \ \text{seconds} \).
Use the formula for linear speed in circular motion: \( v = R_{\text{effective}} \cdot \omega \). Substitute the values of \( R_{\text{effective}} \) and \( \omega \) into this equation to find the speed of the student.
Simplify the expression to calculate the final speed. Ensure all units are consistent (meters and seconds) throughout the calculation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rotational Speed
Rotational speed refers to how fast an object rotates around an axis. For the Earth, this speed varies with latitude; points on the equator move faster than those near the poles due to the Earth's spherical shape. The rotational speed can be calculated using the formula v = ωr, where v is linear speed, ω is angular velocity, and r is the radius of the circular path.
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Latitude and Circumference
Latitude is a geographic coordinate that specifies the north-south position of a point on the Earth's surface. It affects the circumference of the circular path an object follows due to Earth's rotation. At latitude 26 degrees, the effective radius of rotation is smaller than at the equator, leading to a lower linear speed for a person located there.
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Centrifugal Force
Centrifugal force is an apparent force that acts outward on a body moving around a center, arising from the body's inertia. As the Earth rotates, this force affects objects on its surface, including a physics student in Miami. While it does not directly influence the speed calculation, it is important for understanding the effects of Earth's rotation on weight and motion.
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Related Practice
Textbook Question
Peregrine falcons are known for their maneuvering ability. In a tight circular turn, a falcon can attain a centripetal acceleration 1.5 times the free-fall acceleration. What is the radius of the turn if the falcon is flying at 25 m/s?
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
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
The earth's radius is about 4000 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5000 miles. The flight from Kampala to Singapore takes 9.0 hours. What is the plane's angular velocity with respect to the earth's surface? Give your answer in °/h.
Textbook Question
FIGURE EX4.23 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first 4 s?
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