Domanda del libro di testo
A particle moves from A to D in FIGURE EX10.36 while experiencing force F = (6i + 8j) N. How much work does the force do if the particle follows path ACD. Is this a conservative force? Explain.
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Ch 10: Interactions and Potential Energy
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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TOC.tableOfContents
- 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
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Capitolo 10, Problema 34b
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?
Guida verificata passo dopo passo1
To determine the points of equilibrium, calculate the derivative of the potential energy function U(x) with respect to x. This gives the force acting on the particle: F(x) = -dU/dx. For equilibrium, the force must be zero, so solve dU/dx = 0.
Differentiate U(x) = x^3 - 3x with respect to x. This gives dU/dx = 3x^2 - 3. Set this equal to zero to find the equilibrium points: 3x^2 - 3 = 0.
Solve the equation 3x^2 - 3 = 0 for x. Factorize or simplify to find the values of x where equilibrium occurs. This will give the equilibrium points.
To determine the stability of each equilibrium point, calculate the second derivative of U(x), d²U/dx². If d²U/dx² > 0 at a point, it is a point of stable equilibrium. If d²U/dx² < 0, it is a point of unstable equilibrium.
Differentiate dU/dx = 3x^2 - 3 again to find d²U/dx². Evaluate d²U/dx² at each equilibrium point found earlier to classify them as stable or unstable.
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Concetti chiave
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Potential Energy
Potential energy is the energy stored in a system due to its position or configuration. In this case, the potential energy U = x³ - 3x describes how the energy changes as the particle moves along the x-axis. Understanding potential energy is crucial for analyzing the forces acting on the particle and determining its equilibrium points.
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Percorso guidato
07:24
Potential Energy Graphs
Equilibrium Points
Equilibrium points occur where the net force acting on a particle is zero, meaning the particle is in a state of rest or constant motion. These points can be classified as stable or unstable based on the behavior of the potential energy around them. A stable equilibrium is characterized by a local minimum in potential energy, while an unstable equilibrium corresponds to a local maximum.
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10:13Torque & Equilibrium
Stability Analysis
Stability analysis involves examining the second derivative of the potential energy function to determine the nature of equilibrium points. If the second derivative is positive at a point, it indicates a stable equilibrium, as small displacements will return the particle to equilibrium. Conversely, if the second derivative is negative, the point is unstable, as displacements will lead the particle further away from equilibrium.
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Percorso guidato
05:00Dimensional Analysis
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
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?
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Domanda del libro di testo
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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Domanda del libro di testo
How much work is done by the environment in the process shown in FIGURE EX10.39? Is energy transferred from the environment to the system or from the system to the environment?
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Domanda del libro di testo
A cable with 20.0 N of tension pulls straight up on a 1.50 kg block that is initially at rest. What is the block's speed after being lifted 2.00 m? Solve this problem using work and energy.
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