Question

Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. $$\mathbf{F}=e^{x+y}\langle 1,1, z\rangle \text { from } A(0,0,0) \text { to } B(-1,2,-4)$$

Short answer

Answer: The work required to move the object is \((e - 1) - \frac{3}{4}e\), and the force field is non-conservative.

Step-by-step solution

Given Information

The given force field is \(\mathbf{F}=e^{x+y}\langle 1,1, z\rangle\), and we will find the work done in moving an object from point A(0,0,0) to point B(-1,2,-4).

Parameterize the Line Segment

In order to calculate the work along the line segment, we first need a parameterization of the line between points A and B. Let \(\mathbf{r}(t)\) be the parameterization of this line, where \(0 \leq t \leq 1\). Then, $$\mathbf{r}(t) = (1-t)A + tB = (1-t)\langle 0,0,0 \rangle + t\langle -1,2,-4\rangle = \langle -t, 2t, -4t \rangle.$$

Find the Differential of the Parameterization

Next, we need to find the differential of the parameterization, \(d\mathbf{r}\), with respect to the parameter \(t\). We will use this to calculate the work done. Taking the derivative of \(\mathbf{r}(t)\) with respect to \(t\), we get $$\frac{d\mathbf{r}}{dt} = \langle -1, 2, -4 \rangle,$$ so \(d\mathbf{r} = \langle -dt, 2dt, -4dt \rangle\).

Find the Force Acting on the Line

Now, let us find the force acting along the line. We substitute \(x=-t, y=2t, z=-4t\) into the given force field \(\mathbf{F}=e^{x+y}\langle 1,1, z\rangle\): $$\mathbf{F}(t)=e^{t(2-1)}\langle 1,1, -4t\rangle = e^t\langle 1,1, -4t\rangle.$$

Calculate the Work Done

Now, we can calculate the work done by integrating the dot product of \(\mathbf{F}\) and \(d\mathbf{r}\) with respect to \(t\) from 0 to 1: $$W = \int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^1 \mathbf{F}(t) \cdot \langle -dt, 2dt, -4dt \rangle = \int_0^1 e^t( -1 + 2 - 16t^2) dt.$$ $$W = \int_0^1 e^t (-1 + 2 - 16t^2) dt = \int_0^1 e^t (1 - 16t^2) dt,$$ which we integrate and get: $$W = \left[ e^t - \frac{1}{8}e^t(16t^2-2) \right]_0^1 = e - \frac{7}{4}e - 1 + \frac{1}{4} = (e - 1) - \frac{3}{4}e.$$ So the work done is \(W = (e - 1) - \frac{3}{4}e\).

Check if the Force is Conservative

To check if the force is conservative, we need to find the curl of the force field \(\mathbf{F}\). The curl of a force field is given by \(\nabla\times\mathbf{F}\). Here, \(\mathbf{F}=e^{x+y}\langle 1,1, z\rangle\). So, $$\nabla\times\mathbf{F} = \left|\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ e^{x+y} & e^{x+y} & ze^{x+y} \end{matrix}\right|.$$ After calculating the curl, we get: $$\nabla \times\mathbf{F} = \langle 0, ze^{x+y} - e^{x+y}, -e^{x+y} \rangle.$$ The curl \(\nabla \times\mathbf{F}\) is non-zero; hence, the force field is non-conservative. In conclusion, the work required to move an object from point A(0,0,0) to point B(-1,2,-4) in the given force field is \((e - 1) - \frac{3}{4}e\), and the force field is non-conservative.

Key concepts

Force Field

In physics, a force field is a vector field that describes the non-contact forces acting on an object. For a given point in space, the force field provides a vector that represents the force exerted by the field on an object at that point. In this exercise, we have a force field described by the expression \( \mathbf{F}=e^{x+y}\langle 1,1, z\rangle \). This formula indicates that the field's strength and direction change with the coordinates \( x, y, \) and \( z \).
In simpler terms, the field stretches out in all directions from a point and affects objects within its reach without physical contact. This concept is crucial in analyzing how forces act in various physical situations, both in natural phenomena and engineered systems. The force field not being uniform means that the magnitude and direction of the force change across locations.

Parameterization

Parameterization is the process of representing a curve, surface, or line in space using parameters. For instance, we can represent the line from point \( A(0,0,0) \) to point \( B(-1,2,-4) \) using a parameter \( t \) that runs from 0 to 1. This way, any point on the line can be described as \( \mathbf{r}(t) = \langle -t, 2t, -4t \rangle \).
Essentially, parameterization allows us to study the line by focusing on how it behaves within a specific bounded parameter rather than dealing directly with spatial coordinates. It is especially useful when calculating integrals over paths or surfaces, enabling a straightforward approach to solving complex equations.

Line Integral

The concept of a line integral extends the idea of integration to functions that vary along a path. In terms of physics, the line integral of a force field along a curve determines the work done by the force as an object moves along that curve. In this exercise, to compute the work done by the force field \( \mathbf{F} = e^{x+y}\langle 1,1, z\rangle \), we calculate the line integral of the field along the line segment between the points A and B.
Mathematically, this involves integrating the dot product of the vector field and the differential vector of the parameterized path. This dot product reflects how the force aligns with the direction of motion, and integrating it over the interval provides the total work done, capturing the essence of force acting through displacement.

Non-Conservative Force

A non-conservative force is one where the work done by the force field on an object depends on the path taken by the object, not just on the initial and final positions. This characteristic emerges when the curl of a force field is non-zero, indicating that the field is non-conservative. In this case, we examined the force field \( \mathbf{F} = e^{x+y}\langle 1,1, z\rangle \) and found that its curl is non-zero.
Non-conservative forces commonly appear in real-world scenarios, such as friction and air resistance, where energy is dissipated as heat or sound, making recovery of potential energy impossible. They contrast with conservative forces like gravity, where the energy is conserved within a closed system. Understanding whether a force is conservative or non-conservative helps predict energy changes within a system.