Textbook Question
FIGURE EX10.28 shows the potential-energy diagram for a 500 g particle as it moves along the x-axis. Suppose the particle's mechanical energy is 12 J. Where are the particle's turning points?
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Ch 10: Interactions and Potential 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 10, Problem 26b
In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?
Verified step by step guidance1
Step 1: Analyze the graph provided. The graph shows the potential energy U(x) as a function of position x. At point A, the potential energy is approximately 2 J, and at point B, the potential energy is 0 J. The peak of the potential energy curve is around 5 J.
Step 2: Use the principle of conservation of mechanical energy. The total mechanical energy (kinetic energy + potential energy) at point B must be equal to the total mechanical energy at point A for the particle to reach point A.
Step 3: Write the equation for conservation of energy: \( K_B + U_B = K_A + U_A \), where \( K_B \) and \( K_A \) are the kinetic energies at points B and A, and \( U_B \) and \( U_A \) are the potential energies at points B and A.
Step 4: At point A, the particle must have zero kinetic energy to just reach the point (minimum speed condition). Thus, \( K_A = 0 \). Substitute \( U_B = 0 \) and \( U_A = 2 \) J into the equation: \( K_B = U_A - U_B \).
Step 5: Express \( K_B \) in terms of the particle's mass and velocity: \( K_B = \frac{1}{2}mv^2 \). Solve for the velocity \( v \) using \( v = \sqrt{\frac{2(U_A - U_B)}{m}} \), where \( m = 0.1 \) kg (converted from 100 g).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Potential Energy
Potential energy (U) is the energy stored in an object due to its position in a force field, commonly gravitational or elastic. In the context of the graph, it represents the energy of the particle at points A and B, where the height of the curve indicates the potential energy at those positions. Understanding how potential energy changes as the particle moves is crucial for determining the minimum speed required to reach point A.
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Potential Energy Graphs
Conservation of Energy
The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the sum of kinetic energy (KE) and potential energy (PE) at point B must equal the potential energy at point A. This relationship allows us to calculate the minimum speed needed at point B by equating the energies at both points.
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Kinetic Energy
Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. To find the minimum speed at point B, we need to determine how much kinetic energy is required to convert into potential energy as the particle moves to point A. This concept is essential for solving the problem by linking speed to energy.
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Related Practice
Textbook Question
In FIGURE EX10.27, what is the maximum speed of a 2.0 g particle that oscillates between x = 2.0 mm and x = 8.0 mm
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Textbook Question
The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?
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Textbook Question
FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?
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Textbook Question
In FIGURE EX10.28, what is the maximum speed a 200 g particle could have at x = 2.0 m and never reach x = 6.0 m?
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Textbook Question
FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0 m. What is the particle's maximum speed? At what position does it have this speed?
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