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 29
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?
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 x = 2.0 m, U(x) = 0 J, and at x = 6.0 m, U(x) = 4 J. The particle's total mechanical energy must be less than or equal to 4 J to ensure it does not reach x = 6.0 m.
Step 2: Use the principle of conservation of mechanical energy. The total mechanical energy E is the sum of the kinetic energy K and potential energy U. At x = 2.0 m, the potential energy U is 0 J, so the total energy E is equal to the kinetic energy K at this position.
Step 3: Write the expression for kinetic energy K: \( K = \frac{1}{2} m v^2 \), where m is the mass of the particle and v is its speed. The mass of the particle is given as 200 g, which should be converted to kilograms: \( m = 0.2 \, \text{kg} \).
Step 4: Set the total energy E equal to the maximum allowable energy (4 J) to ensure the particle does not reach x = 6.0 m. Solve for the maximum speed v using \( E = \frac{1}{2} m v^2 \). Rearrange the equation to find \( v = \sqrt{\frac{2E}{m}} \).
Step 5: Substitute the values for E (4 J) and m (0.2 kg) into the equation \( v = \sqrt{\frac{2E}{m}} \). This will give the maximum speed the particle can have at x = 2.0 m without reaching x = 6.0 m.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Potential Energy (U)
Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. In the context of the graph, it represents the energy of the particle at various positions along the x-axis. The height of the curve indicates the potential energy at each position, which influences the particle's ability to move to different locations.
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Gravitational Potential Energy
Conservation of Energy
The principle of conservation of energy states that the total energy in a closed system remains constant. For the particle in the problem, the sum of its kinetic energy and potential energy must equal a constant value. This means that as the particle moves, any change in potential energy will result in a corresponding change in kinetic energy, affecting its speed at different positions.
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Kinetic Energy (KE)
Kinetic energy is the energy of an object due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In this problem, the maximum speed of the particle at x = 2.0 m can be determined by considering the potential energy at that point and ensuring that the particle has enough kinetic energy to not reach x = 6.0 m, where the potential energy is higher.
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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
A system in which only one particle moves has the potential energy shown in FIGURE EX10.31. What is the x-component of the force on the particle at x = 5, 15, and 25 cm?
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Textbook Question
A particle moving along the y-axis is in a system with potential energy U = 4y3 J, where y is in m. What is the y-component of the force on the particle at y = 0 m, 1 m, and 2 m?
Textbook Question
In FIGURE EX10.26, What minimum speed does a 100 g particle need at point B to reach point A?
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Textbook Question
A particle moving along the x-axis is in a system that has potential energy U = x3 - 3x J, where x is in m. For each, is it a point of stable or unstable equilibrium?