91Ó°ÊÓ

A force field is a vector field that describes the forces acting on every point in a space. In this context, the force field is given by \( \mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} - 5z \mathbf{k} \). This equation tells us how the force varies depending on the position in three-dimensional space.

When you have a force field:

• The first part, \(x \mathbf{i}\), determines the force's influence in the x-direction, which is proportional to x.
• The \(y \mathbf{j}\) component describes how the force affects motion along the y-direction, also directly linked to the y-coordinate.
• The term \(-5z \mathbf{k}\) indicates that the force in the z-direction is opposite to the increase in z, and it scales linearly with \(-5z\).

Understanding the force field is crucial for analyzing how any object moving within it will behave. It provides a general map of how the forces will act, influencing how we calculate the work done and movement along any specified path.

A parameterized path is a way to describe a curve in space using a parameter, typically denoted as \(t\). The path is defined with functions describing its x, y, and z coordinates in terms of \(t\).

In this exercise, the path \(C\) is parameterized by:

• \( \mathbf{r}(t) = 2 \cos t \mathbf{i} \) for the x-coordinate, shaping a circular path in the x-direction.
• \( \mathbf{r}(t) = 2 \sin t \mathbf{j} \) for the y-coordinate, maintaining circular symmetry in the y-direction.
• \( \mathbf{r}(t) = t \mathbf{k} \) for the z-coordinate, representing a linear change as \(t\) increases, creating a helical path.

This path traverses a helix - a common shape that spirals upwards or downwards as \(t\) progresses from \(0\) to \(2\pi\).

• For circular paths, the radius in the xy-plane is constant, determined by the coefficients of \(\cos t\) and \(\sin t\).
• The linear term in the z-component causes the object to rise or fall as it revolves around the circle, resembling a staircase in the air.

Understanding a parameterized path helps us locate every point along the movement trajectory, which is key to computing the work done by the force field.

The concept of work done by a force field is closely linked to the idea of integrating force along a path. When an object moves through a force field, the environment exerts forces that require or do work.

The mathematical expression for calculating the work done by a force as it acts along a path is given by the line integral: \[ W = \int_C \mathbf{F} \cdot d\mathbf{r} \]Here, the line integral sums up the force's influence over each small segment \(d\mathbf{r}\) of the path \(C\). Notably, this exercise involves calculating the dot product:

• The dot product \( \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \) translates the force components relative to their influence on the path's direction and magnitude.
• The negative sign in the scalar result \(-5t\) indicates a directionally opposing or energetically resisting movement in the z-direction along our parameterized helix.

Integrating this result over the full cycle from \(t = 0\) to \(t = 2\pi\) provides the total work done, resulting in a value of \(-10\pi^2\). This negative result implies that the force field takes energy from the object, resisting its motion along the helical path.