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. $$\mathbf{F}=\langle x, y\rangle \text { from } A(1,1) \text { to } B(3,-6)$$

Short answer

Question: Calculate the work required to move an object in the force field \(\mathbf{F}(x, y) = \langle x, y \rangle\) between points A(1, 1) and B(3, -6) and determine whether the force field is conservative or not. Answer: The work required to move the object between points A and B is \(\frac{13}{2}\) and the force field is conservative.

Step-by-step solution

Parametrize the Path

To parametrize the path between points A(1,1) and B(3,-6), consider a straight line segment connecting the two points. The parametrization of this line segment can be defined by a vector-valued function \(\mathbf{r}(t) = \langle x(t), y(t) \rangle\): $$x(t) = 1 + 2t$$ $$y(t) = 1 - 7t$$ where \(t\) ranges from 0 to 1.

Compute the Line Integral of the Force Field

To compute the line integral of the force, we need to calculate the dot product of the force field and the derivative of the parametrization vector, and then integrate it over the path: $$\text{Work} = \int_{0}^{1} \mathbf{F}(x(t), y(t)) \cdot \frac{d\mathbf{r}}{dt} dt$$ First, calculate the derivative of the parametrization vector \(\frac{d\mathbf{r}}{dt} = \langle \frac{dx}{dt}, \frac{dy}{dt} \rangle\): $$\frac{dx}{dt} = 2$$ $$\frac{dy}{dt} = -7$$ So, \(\frac{d\mathbf{r}}{dt} = \langle 2, -7 \rangle\). Now, evaluate the force field at the parametrized path \(\mathbf{F}(x(t), y(t)) = \langle x(t), y(t) \rangle\): $$\mathbf{F}(x(t), y(t)) = \langle 1 + 2t, 1 - 7t \rangle$$ Compute the dot product of the force field and the derivative of the parametrization vector: $$(\mathbf{F}(x(t), y(t)) \cdot \frac{d\mathbf{r}}{dt}) = (1 + 2t)(2) + (1 - 7t)(-7) = 17t - 2$$ Finally, integrate the dot product of the force field and the derivative of the parametrization vector from 0 to 1: $$\text{Work} = \int_{0}^{1} (17t - 2) dt = \left[\frac{17}{2}t^2 - 2t \right]_0^1 = \frac{17}{2} - 2 = \frac{13}{2}$$ Thus, the work required to move the object along the line segment between points A and B is \(\frac{13}{2}\).

Determine if the Force Field is Conservative

To check if the force field is conservative, we need to compute its curl (the cross product of the gradient and the force field). If the curl is zero, then the force field is conservative: $$\text{Curl}(\mathbf{F}) = \nabla \times \mathbf{F} = \frac{\partial y}{\partial x} - \frac{\partial x}{\partial y} = 0 - 0 = 0$$ Since the curl is zero, the force field is conservative.

Key concepts

Line Integral

A line integral is a mathematical tool used to calculate the work done by a force field on a moving particle along a path. It's like summing up the skill and energy needed at every little step of a journey.
In physics, the line integral is represented as the integral of the dot product of the force field and the path's differential element. This product essentially tells you how much of the force is actually being used to move from point A to point B.
Understanding line integrals is crucial as they allow us to work with vector fields and curves rather than simple linear paths. They help in determining the work done in scenarios involving varying paths or forces.

• The main idea is to break down the path into very tiny segments.
• Calculate the force contribution along each segment.
• Sum these contributions for the entire path, resulting in the total work done.

Parametrization

Parametrization is the process of defining a path or curve using a parameter, usually denoted as \( t \). It's like giving your path a formula that describes every point along it, kind of like drawing a trail using a mathematical pen.
This method is particularly helpful because instead of dealing with multiple equations for \( x \) and \( y \), you have a single parameter \( t \) controlling both. In our exercise, the curve between points \((1,1)\) and \((3,-6)\) is described by:

• \( x(t) = 1 + 2t \)
• \( y(t) = 1 - 7t \)

Here, \( t \) ranges from 0 to 1, which represents the start and end of the journey, respectively.
Parametrization helps in simplifying calculations, enabling a clear understanding of how we traverse from start to end, crucial for calculating line integrals.

Conservative Force

A conservative force is one where the work done in moving an object between two points is independent of the path taken. Forces like gravity are conservative because they only depend on the starting and ending positions, not the path.
To determine whether a force field is conservative, we typically check the curl. If the curl is zero throughout the field, then it's conservative. This essentially means there's no rotational component skewing the force's effect, like pulling on a rope rather than twisting it.
In practical terms, recognizing a conservative force field is important because it allows us to use potential energy to simplify our calculations by focusing only on the initial and final positions.

• If conservative, total mechanical energy is conserved.
• Path independence simplifies work and energy calculations.

Curl of a Vector Field

The curl of a vector field is a measure of how much the field tends to rotate around a point. It helps you see the twisty, swirling nature of the field, akin to how water might swirl down a drain.
Mathematically, it's defined using a cross product involving the gradient operator. For a field \( \mathbf{F} = \langle P, Q \rangle \), the curl is given by \( abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).
For the field \( \langle x, y \rangle \), the calculation simplifies to zero:

• \( \frac{\partial y}{\partial x} = 0 \)
• \( \frac{\partial x}{\partial y} = 0 \)
• Thus, \( abla \times \mathbf{F} = 0 \)

No curl means no swirling, indicating the force field is conservative. Understanding the curl is essential as it helps in identifying conservative fields and applications in electromagnetism and fluid dynamics.