Textbook Question
Doug uses a 25 N horizontal force to push a 5.0 kg crate up a 2.0-m-high, 20° frictionless slope. What is the speed of the crate at the top of the slope?
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Ch 09: Work and Kinetic Energy
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 9, Problem 46
Susan's 10 kg baby brother Paul sits on a mat. Susan pulls the mat across the floor using a rope that is angled 30° above the floor. The tension is a constant 30 N and the coefficient of friction is 0.20. Use work and energy to find Paul's speed after being pulled 3.0 m.
Verified step by step guidance1
Identify the forces acting on the system: The forces include the tension in the rope (30 N), the gravitational force on Paul (mg = 10 kg × 9.8 m/s²), the normal force, and the frictional force. The tension has a vertical component and a horizontal component due to the 30° angle.
Break the tension force into components: The horizontal component of the tension is Tₓ = T × cos(30°), and the vertical component is Tᵧ = T × sin(30°). Use these components to analyze the motion.
Calculate the work done by each force: The work done by the tension force is Wₜ = Tₓ × d, where d = 3.0 m. The work done by friction is Wₓ = -f × d, where f = μ × N (frictional force) and N = mg - Tᵧ (adjusted normal force).
Apply the work-energy principle: The net work done on Paul is equal to the change in kinetic energy, ΔKE = Wₜ + Wₓ. Since Paul starts from rest, his initial kinetic energy is 0, so ΔKE = 0.5 × m × v², where v is the final speed.
Solve for the final speed: Rearrange the work-energy equation to solve for v. Substitute the expressions for Wₜ and Wₓ, and calculate the net work. Then, use v = √((2 × net work) / m) to find Paul's speed after being pulled 3.0 m.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 scenario, the work done by the tension in the rope will contribute to increasing Paul's kinetic energy as he is pulled across the mat. Understanding this principle allows us to relate the forces acting on Paul to his resulting speed.
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The Work-Energy Theorem
Friction and Normal Force
Friction is a force that opposes motion between two surfaces in contact. The coefficient of friction quantifies this resistance. In this case, the normal force acting on Paul, which is affected by the weight of the baby and the angle of the rope, will determine the frictional force opposing his motion. Calculating the net force requires understanding how these forces interact.
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Tension in a Rope
Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In this problem, the tension of 30 N at an angle of 30° affects both the horizontal and vertical components of the force acting on Paul. Analyzing the components of tension is crucial for determining the net force and subsequently the acceleration of Paul.
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Related Practice
Textbook Question
A ball shot straight up with kinetic energy K₀ reaches height h. What height will it reach if the initial kinetic energy is doubled?
Textbook Question
A 150 g particle at x = 0 is moving at 2.00 m/s in the + x - direction. As it moves, it experiences a force given by Fₓ = (0.250 N) sin (x/2.00 m). What is the particle's speed when it reaches x = 3.14 m?
Textbook Question
A 1000 kg elevator accelerates upward at 1.0 m/s² for 10 m, starting from rest. How much work does the tension in the elevator cable do on the elevator?
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Textbook Question
A pile driver lifts a 250 kg weight and then lets it fall onto the end of a steel pipe that needs to be driven into the ground. A fall from an initial height of 1.5 m drives the pipe in 35 cm. What is the average force that the weight exerts on the pipe?
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Textbook Question
A 50 kg ice skater is gliding along the ice, heading due north at 4.0 m/s. The ice has a small coefficient of static friction, to prevent the skater from slipping sideways, but μk = 0. Suddenly, a wind from the northeast exerts a force of 4.0 N on the skater. Use work and energy to find the skater's speed after gliding 100 m in this wind.