Textbook Question
A 4.0-cm-diameter, 55 g ball is shot horizontally into a tank of 40Β°C honey. How long will it take for the horizontal speed to decrease to 10% of its initial value?
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Ch 06: Dynamics I: Motion Along a Line
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 6, Problem 74b
A block of mass m is at rest at the origin at t = 0. It is pushed with constant force Fβ from π = 0 to π = L across a horizontal surface whose coefficient of kinetic friction is ΞΌβ = ΞΌβ ( 1 - π/L ) . That is, the coefficient of friction decreases from ΞΌβ at π = 0 to zero at π = L. b. Find an expression for the block's speed as it reaches position L.
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Step 1: Begin by identifying the forces acting on the block. The block experiences a constant applied force Fβ, a frictional force that depends on position π, and the normal force balancing the gravitational force. The frictional force is given by F_friction = ΞΌβ * m * g, where ΞΌβ = ΞΌβ * (1 - π/L).
Step 2: Write the net force acting on the block as a function of position π. The net force is F_net = Fβ - F_friction. Substituting the expression for F_friction, we get F_net = Fβ - ΞΌβ * (1 - π/L) * m * g.
Step 3: Use Newton's second law to relate the net force to the block's acceleration. Newton's second law states F_net = m * a, where a is the acceleration. Substituting F_net, we have m * a = Fβ - ΞΌβ * (1 - π/L) * m * g. Simplify to find the acceleration: a = (Fβ/m) - ΞΌβ * g * (1 - π/L).
Step 4: To find the block's speed at position π = L, use the work-energy principle. The work done by the net force is equal to the change in kinetic energy. The work done by the applied force and friction can be calculated by integrating the net force over the distance from π = 0 to π = L. The work-energy principle states: W_net = ΞK = (1/2) * m * vΒ² - 0, where v is the speed at π = L.
Step 5: Perform the integration to find the work done by the net force. The net force is F_net = Fβ - ΞΌβ * (1 - π/L) * m * g. Integrate F_net with respect to π from 0 to L: W_net = β«[0 to L] (Fβ - ΞΌβ * (1 - π/L) * m * g) dπ. Solve this integral to find W_net, then equate it to (1/2) * m * vΒ² to solve for v, the block's speed at position L.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Newton's Second Law of Motion
Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing the motion of the block under the influence of the applied force and friction.
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Intro to Forces & Newton's Second Law
Friction and its Coefficient
Friction is the force that opposes the relative motion of two surfaces in contact. The coefficient of kinetic friction (ΞΌβ) quantifies this force and varies depending on the surfaces involved. In this scenario, the coefficient decreases linearly from ΞΌβ to zero as the block moves from position 0 to L, affecting the net force and, consequently, the block's acceleration and speed.
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Work-Energy Principle
The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this problem, the work done by the applied force and the work done against friction will determine the block's final speed at position L. By calculating the net work done on the block, one can derive an expression for its speed as it reaches the end of the distance L.
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Related Practice
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