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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

Expert verified
Force (a) is conservative with potential energy \( U = -kx^2/2 - ky^2 - 3kz^2/2 \).

Step by step solution

01

Understand Conservative Forces

A force is conservative if the work done by the force around any closed path is zero or, equivalently if it can be expressed as the gradient of a scalar potential function.
02

Check for Zero Curl

For a force \( \mathbf{F} = (F_x, F_y, F_z) \) to be conservative, its curl should be zero, i.e., \( abla \times \mathbf{F} = 0 \). Calculate the curl for each force.
03

Calculate Curl for Each Force (a)

For \( \mathbf{F} = k(x, 2y, 3z) \), the curl is \( abla \times \mathbf{F} = (0, 0, 0) \), which is zero. Thus, force (a) is conservative.
04

Calculate Curl for Force (b)

For \( \mathbf{F} = k(y, x, 0) \), the curl is \( abla \times \mathbf{F} = (0, 0, -k) \), which is non-zero. Thus, force (b) is not conservative.
05

Calculate Curl for Force (c)

For \( \mathbf{F} = k(-y, x, 0) \), the curl is \( abla \times \mathbf{F} = (0, 0, 2k) \), which is non-zero. Thus, force (c) is not conservative.
06

Find Potential Energy for Force (a)

Since force (a) is conservative, we can find potential energy \( U \) by integrating: \[ U = - \int{kx dx} - \int{2ky dy} - \int{3kz dz} = -kx^2/2 - ky^2 - 3kz^2/2 \]
07

Verify Force Equation

To verify, differentiate the potential energy function: \( abla U = (kx, 2ky, 3kz) \). Thus, \( \mathbf{F} = -abla U \) confirms our force \( \mathbf{F} = k(x, 2y, 3z) \) is correctly represented by its potential energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
Potential energy is an important concept in physics, representing the energy stored in a system due to the position or configuration of its components. This energy can be transformed into kinetic energy or other forms of energy, allowing a system to do work. In the context of conservative forces, the potential energy is related to how forces act within the system.

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
The gradient is a vector operation that describes the rate and direction of change in a scalar field. Think of a hill: the gradient at any point on the hill points in the direction of the steepest ascent and its magnitude tells you how steep the hill is.

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
Curl is a vector operation used to determine the rotation or 'twist' of a vector field in three-dimensional space. When you find the curl of a vector field, you get another vector that indicates the axis of rotation and the magnitude of the rotation in that field.

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.

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Most popular questions from this chapter

Consider a head-on elastic collision between two particles. (since the collision is head-on, the motion is confined to a single straight line and is therefore one-dimensional.) Prove that the relative velocity after the collision is equal and opposite to that before. That is, \(v_{1}-v_{2}=-\left(v_{1}^{\prime}-v_{2}^{\prime}\right),\) where \(v_{1}\) and \(v_{2}\) are the initial velocities and \(v_{1}^{\prime}\) and \(v_{2}^{\prime}\) the corresponding final velocities.

Consider a small frictionless puck perched at the top of a fixed sphere of radius \(R\). If the puck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere? [Hint: Use conservation of energy to find the puck's speed as a function of its height, then use Newton's second law to find the normal force of the sphere on the puck. At what value of this normal force does the puck leave the sphere?]

Consider the Atwood machine of Figure \(4.15,\) but suppose that the pulley has radius \(R\) and moment of inertia \(I\). (a) Write down the total energy of the two masses and the pulley in terms of the coordinate \(x\) and \(\dot{x}\). (Remember that the kinetic energy of a spinning wheel is \(\frac{1}{2} I \omega^{2}\).) (b) Show (what is true for any conservative one-dimensional system) that you can obtain the equation of motion for the coordinate \(x\) by differentiating the equation \(E=\) const. Check that the equation of motion is the same as you would obtain by applying Newton's second law separately to the two masses and the pulley, and then eliminating the two unknown tensions from the three resulting equations.

Both the Coulomb and gravitational forces lead to potential energies of the form \(U=\gamma / | \mathbf{r}_{1}-\) \(\mathbf{r}_{2} |,\) where \(\gamma\) denotes \(k q_{1} q_{2}\) in the case of the Coulomb force and \(-G m_{1} m_{2}\) for gravity, and \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are the positions of the two particles. Show in detail that \(-\nabla_{1} U\) is the force on particle 1 and \(-\nabla_{2} U\) that on particle 2.

Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z):(\mathbf{a}) f=\ln (r),\) (b) \(f=r^{n}\), (c) \(f=g(r),\) where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(g(r)\) is some unspecified function of \(r .\) [Hint: Use the chain rule.]

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