Question

Given the force field \(\mathbf{F},\) find the work required to move an object on the given oriented curve. \(\mathbf{F}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}\) on the line segment from (1,1,1) to (8,4,2)

Short answer

Answer: To find the work done, we need to evaluate the line integral \(W = \int_{0}^{1} \mathbf{F}(t) \cdot \frac{d\mathbf{r}}{dt} \, dt\), where \(\mathbf{F}(t) = \frac{\langle 1 + 7t, 1 + 3t, 1 + t\rangle}{(1+7t)^2 + (1+3t)^2 + (1+t)^2}\) and \(\frac{d\mathbf{r}}{dt} = \langle 7, 3, 1 \rangle\). The work done is obtained by evaluating this integral using numerical methods or software.

Step-by-step solution

Parametrize the line segment

To parametrize the line segment from the point A(1,1,1) to the point B(8,4,2), we can use the vector equation of a line that connects A and B. \(\mathbf{r}(t) = \mathbf{A} + t (\mathbf{B} - \mathbf{A})\), where \(0 \leq t \leq 1\) In this case, \(\mathbf{A} = \langle 1, 1, 1 \rangle\) and \(\mathbf{B} = \langle 8, 4, 2 \rangle\). Hence, the equation becomes: \(\mathbf{r}(t) = \langle 1, 1, 1 \rangle + t (\langle 8, 4, 2 \rangle - \langle 1, 1, 1 \rangle) = \langle 1, 1, 1 \rangle + t \langle 7, 3, 1 \rangle\) Now, we have the parametrization of the line segment: \(\mathbf{r}(t) = \langle 1 + 7t, 1 + 3t, 1 + t \rangle\)

Calculate the vector field on the line segment.

First, we need to find the vector field \(\mathbf{F}(t) = \frac{\langle x(t), y(t), z(t)\rangle}{x^2(t)+y^2(t)+z^2(t)}\), where x(t) = 1 + 7t, y(t) = 1 + 3t, and z(t) = 1 + t. \(\mathbf{F}(t) = \frac{\langle 1 + 7t, 1 + 3t, 1 + t\rangle}{(1+7t)^2 + (1+3t)^2 + (1+t)^2}\)

Find the derivative of the position vector r(t).

To find the line integral, we need to find the derivative of the position vector \(\mathbf{r}(t) = \langle 1 + 7t, 1 + 3t, 1 + t \rangle\) with respect to t. \(\frac{d\mathbf{r}}{dt} = \langle 7, 3, 1 \rangle\)

Compute the line integral of the vector field along the path.

To compute the line integral of the vector field \(\mathbf{F}(t)\) along the path of the line segment, we use the following formula: \(W = \int_{0}^{1} \mathbf{F}(t) \cdot \frac{d\mathbf{r}}{dt} \, dt\) \(W = \int_{0}^{1} \left(\frac{\langle 1 + 7t, 1 + 3t, 1 + t\rangle}{(1+7t)^2 + (1+3t)^2 + (1+t)^2}\right) \cdot \langle 7, 3, 1 \rangle \, dt\) Evaluate this integral to find the work done.

Evaluate the integral for work done

We can now evaluate the line integral: \(W = \int_{0}^{1} \left(\frac{\langle 1 + 7t, 1 + 3t, 1 + t\rangle}{(1+7t)^2 + (1+3t)^2 + (1+t)^2}\right) \cdot \langle 7, 3, 1 \rangle \, dt\) We will use numerical methods or software to calculate the value of this integral to obtain the work done. By evaluating the integral, we find the work done, which represents the work required to move an object on the given oriented curve in the force field \(\mathbf{F}\).

Key concepts

Vector Calculus

Vector Calculus plays a crucial role in understanding how quantities relate in multi-dimensional spaces. It extends our knowledge from simple calculus into three dimensions, allowing us to analyze curves, surfaces, and solids. One important application is finding the work done by a force field along a path, using what’s known as a **line integral**.

This process involves:

• Calculating the path or curve through space.
• Integrating the force applied along this path to find the total work done.

Line integrals combine both directional and vector components, requiring vectors for both force fields and paths. With vector calculus, these can be analyzed in terms of parametric representations and derivatives to compute the integral accurately.

Parametric Equations

Parametric equations provide a method to describe curves using parameters, often represented by a variable like \(t\).

Instead of expressing coordinates such as \(x\) and \(y\) directly as functions of each other, we use a parameter to manage each component of the curve. For instance, given points A and B, the parametric equations are developed to represent the line segment:

\[ \mathbf{r}(t) = \langle 1 + 7t, 1 + 3t, 1 + t \rangle \]

Here's how it works:

• **Initial Point:** Starting at (1, 1, 1).
• **Direction Vector:** Moving toward (8, 4, 2) gives direction \(\langle 7, 3, 1 \rangle\).
• **Parameter Range:** \(0 \leq t \leq 1\), which moves from the start to the end.

With parametric equations, we can easily calculate derivatives, evaluate line integrals, and explore curves elegantly within force fields.

Force Fields

A force field is a vector field that represents the distribution of forces in a certain region. In this exercise, the force field \( \mathbf{F} = \frac{\langle x, y, z \rangle}{x^{2}+y^{2}+z^{2}} \) signifies an inverse-square relationship, commonly seen in physics.

This type of field can model gravitational or electrostatic forces, where the power diminishes as you move away from the source. Understanding a force field involves:

• **Analyzing its influence** on objects within its range.
• **Evaluating the work done** by moving an object along a path through this field, which can be achieved by computing the line integral.

Providing a vector for each point in space, force fields express how forces might act, helping to solve real-world problems in engineering and physics.