Problem 30 - The Calculus with Analytic Geometry

Question

In Exercises 21 through 34 , find the total work done in moving an object along the given arc \(C\) if the motion is caused by the given force field. Assume the arc is measured in inches and the force is measured in pounds. \(F(x, y, z)=z^{2} i+y^{2} j+x z k ; C:\) the line segment from the origin to the point \((4,0,3) .\)

Step-by-Step Solution

Answer

24 inch-pounds

Step 1 - Understand the problem

We need to find the total work done by the given force field \(F(x, y, z) = z^2 i + y^2 j + xz k\) along the line segment from the origin (0, 0, 0) to the point (4, 0, 3).

Step 2 - Parameterize the line segment

To parameterize the line segment, use the vector equation \( \textbf{r}(t) = (1 - t) \textbf{r}_0 + t \textbf{r}_1 \), where \(0 \leq t \leq 1\), \( \textbf{r}_0 = (0, 0, 0) \) and \( \textbf{r}_1 = (4, 0, 3) \). Thus, \( \textbf{r}(t) = (4t, 0, 3t) \).

Step 3 - Compute the derivative of the parameterization

Find \( \textbf{r}'(t) \): \( \textbf{r}'(t) = \frac{d}{dt}(4t, 0, 3t) = (4, 0, 3) \).

Step 4 - Express the force field along the parameterized path

Substitute \( \textbf{r}(t) \) into \( F(x, y, z) \): \( F(4t, 0, 3t) = (3t)^2 i + (0)^2 j + (4t)(3t) k = 9t^2 i + 0 j + 12t^2 k \).

Step 5 - Compute the dot product of \( F(\textbf{r}(t)) \) and \( \textbf{r}'(t) \)

\( F(\textbf{r}(t)) \cdot \textbf{r}'(t) = (9t^2, 0, 12t^2) \cdot (4, 0, 3) = 9t^2 \cdot 4 + 12t^2 \cdot 3 = 36t^2 + 36t^2 = 72t^2 \).

Step 6 - Integrate the dot product from \( t = 0 \) to \( t = 1 \)

Now integrate \( 72t^2 \) from 0 to 1: \[ \int_0^1 72t^2 \ dt = 72 \int_0^1 t^2 \ dt = 72 [\frac{t^3}{3}]_0^1 = 72 (\frac{1}{3} - 0) = 24 \].

Key Concepts

Vector Calculus

Vector calculus is a broad field of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. Important concepts include vector fields, gradient, divergence, and curl.
In this problem, we are given a force field, which is a type of vector field. A force field \(F(x, y, z) = z^2 i + y^2 j + xz k\) associates a force vector with every point in space.
Understanding how vector calculus helps in physics is crucial because it bridges the mathematical representation of physical phenomena, like force fields and fluid flow.

• Vector fields: A vector field assigns a vector to every point in space.
• Gradient: A measurement of how a function changes at any point in space.
• Divergence: Measures the 'outflow' of a vector field from a point.
• Curl: Measures the 'rotation' of a vector field around a point.

By using vector calculus, we can better understand and calculate physical properties such as the work done by a force field, as seen in this example.

Line Integrals

Line integrals are an essential concept in vector calculus. They allow us to integrate a function along a specified curve or path.
In this exercise, we need to compute the total work done by a force field along a line segment.

• Path or curve: The path taken is critical in line integrals. Here, it's a straight line from the origin to the point \( (4, 0, 3)\).
• Force field along the path: By substituting the parameterized coordinates into the force field equation, we find how the force behaves along the path.

We can compute the work done by the field along this segment by performing a dot product of the force vector and the path direction vector (parameterization derivative), eventually integrating this dot product over the length of the path.
The final step involves integrating \( 72t^2 \) from \( t = 0 \) to \( t = 1\) to determine the total work.

Parameterization

Parameterization is a method of expressing a curve or path using a parameter (usually denoted as t). It is very useful in performing line integrals because it provides a simple way to describe complex paths.
For our problem, we parameterize the line segment from the origin \( (0,0,0) \) to the point \( (4,0,3) \). The parameterization can be written as \( \textbf{r}(t) = (4t, 0, 3t) \) where \( 0 \leq t \leq 1 \).

• Parameterize the path: Express each variable (x, y, z) as functions of t.
• Compute derivatives: Calculate the derivatives of the parameterized path, representing direction and magnitude.

Once parameterized, the motion along the line can be smoothly controlled from starting point to end point. This simplification allows us to substitute into the force field equation and perform further operations like dot products and integrations efficiently.