Chapter 4: Problem 23
Which of the following forces is conservative? (a) \(\mathbf{F}=k(x, 2 y, 3 z)\) where \(k\) is a constant. (b) \(\mathbf{F}=k(y, x, 0) .\) (c) \(\mathbf{F}=k(-y, x, 0)\). For those which are conservative, find the corresponding potential energy \(U,\) and verify by direct differentiation that \(\mathbf{F}=-\nabla U\).
Short Answer
Step by step solution
Understand Conservative Forces
Check for Zero Curl
Calculate Curl for Each Force (a)
Calculate Curl for Force (b)
Calculate Curl for Force (c)
Find Potential Energy for Force (a)
Verify Force Equation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Potential Energy
When a force is conservative, there exists a potential energy function, often denoted as \( U \), such that the force can be derived as the negative gradient of this potential energy. This is expressed mathematically as \( \mathbf{F} = -abla U \).
For example, in our problem exercise, the force \( \mathbf{F} = k(x, 2y, 3z) \) is conservative, meaning we can find its potential energy by integrating the components of the force. The potential energy function obtained, \( U = -\frac{kx^2}{2} - ky^2 - \frac{3kz^2}{2} \), shows how the energy is distributed with respect to \( x \), \( y \), and \( z \) in this 3-dimensional system.
By understanding potential energy, we can predict how movement or changes in configuration can affect a system's behavior, making it a fundamental concept in physics.
Gradient
For a scalar field \( U(x, y, z) \), the gradient is denoted by \( abla U \) and results in a vector \((\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z})\). It indicates how the potential energy changes with position in the space.
In the exercise, when we computed \( abla U \) for the potential energy function \( U = -\frac{kx^2}{2} - ky^2 - \frac{3kz^2}{2} \), we found that \( abla U = (kx, 2ky, 3kz) \). This confirms the conservative force \( \mathbf{F} = -abla U \).
Understanding gradients is crucial, as they give insight into how forces will act on particles, pushing them along the path of decreasing potential energy in a conservative field.
Curl
For a force field to be conservative, its curl must be zero, \( abla \times \mathbf{F} = 0 \). This implies that the path taken between two points does not affect the work done, characteristic of conservative fields.
In the exercise, we calculated the curl for several force fields to determine their nature. The force \( \mathbf{F} = k(x, 2y, 3z) \) showed a curl of zero, confirming it as conservative. However, forces \( \mathbf{F} = k(y, x, 0) \) and \( \mathbf{F} = k(-y, x, 0) \) had non-zero curls, making them non-conservative.
Knowing about curl helps us understand the behavior of force fields. It's essential in physics when determining whether certain forces will conserve mechanical energy and how they will influence particle motion. Understanding curl aids in visualizing how fields behave in space, especially in electromagnetism and fluid dynamics.