Question

The Spiral of Archimedes. The spiral of Archimedes is a curve described in polar coordinates by the equation $$ r=k \theta $$ where \(r\) is the distance of a point from the origin, and \(\theta\) is the angle of that point in radians with respect to the origin. Plot the spiral of Archimedes for \(0 \leq \theta \leq 6 \pi\) when \(k=0.5\). Be sure to label your plot properly.

Short answer

To plot the Spiral of Archimedes with \(k = 0.5\) and \(\theta\) ranging from \(0\) to \(6\pi\), use the equation \(r = 0.5\theta\) in a polar coordinate system. The graph will show a spiral starting at the origin and expanding outward, with the distance from the origin increasing by half the angle in radians. Make sure to label the graph with the appropriate equation, range, and value of \(k\).

Step-by-step solution

Understand the Spiral of Archimedes

The Spiral of Archimedes is a mathematical curve defined by a simple polar coordinate equation: \(r = k\theta\), where \(r\) is the distance of a point from the origin, and \(\theta\) is the angle of that point in radians with respect to the origin. The constant \(k\) determines how tightly or loosely the spiral is wound.

Parameters and Range

In this exercise, we are asked to plot the spiral for \(0 \leq \theta \leq 6\pi\) when \(k = 0.5\). So, we will use the equation \(r = 0.5\theta\), and plot the spiral within the specified range of \(\theta\).

Plot the Spiral

To plot the spiral, we will use polar coordinates. In most programming or graphing tools, you can choose a polar coordinate system to plot the curve. Simply input the equation \(r = 0.5\theta\) and set the range for \(\theta\) as \(0 \leq \theta \leq 6\pi\). Once you have plotted the spiral, make sure to label your graph properly and indicate the equation of the spiral, the range of \(\theta\), and the value of \(k\). The resulting graph should show a spiral that starts at the origin (center) and moves outward as \(\theta\) increases, with the distance from the origin increasing by \(0.5\) times the angle in radians.

Key concepts

Polar Coordinates

When you picture a standard graph, you most likely think of the Cartesian coordinate system, with its horizontal (x-axis) and vertical (y-axis) grid. However, polar coordinates offer an alternative method of mapping points, which is essential in certain mathematical contexts, especially when dealing with curves that radiate from a point.

Polar coordinates describe a point’s location based on its distance from a central point (known as the origin or pole) and an angle from a reference direction (commonly, the positive x-axis). Each point is represented as \(r, \theta\), where \(r\) is the radius or the distance to the point from the origin, and \(\theta\) is the angle in radians between the positive x-axis and the point, measured counterclockwise.

This system is particularly useful when plotting circular or spiral shapes because it aligns with the curve's natural symmetry. When we describe the Spiral of Archimedes, we’re praising polar coordinates for their ability to capture the essence of the spiral in such a simple equation: \(r=k\theta\).

MATLAB Plotting

MATLAB is renowned for its powerful visualization capabilities, particularly when it comes to plotting mathematical curves. To plot a curve in MATLAB, you begin by defining the range of your variables; in this case, \(\theta\) from 0 to \(6\pi\). Then, by using built-in functions for polar plotting, you input the polar equation that defines your curve.

In MATLAB, plotting the Spiral of Archimedes involves invoking the 'polarplot' function with your \(\theta\) range and corresponding \(r\) values. MATLAB conveniently takes care of the rest, transforming your equation into a visual representation on a polar coordinate grid.

It's crucial to label your plot correctly to provide context and understanding – this includes titles, legends, and annotations for the radius and angle. With MATLAB’s rich graphics, you can interactively zoom, pan, or even animate plots, offering a deeper insight into the curves you’re studying.

Mathematical Curves

Mathematical curves like the Spiral of Archimedes aren’t just pretty to look at – they encapsulate mathematical relationships and have applications across various fields, from physics to engineering to biology. For instance, the Archimedean spiral serves as a basic model for phenomena such as springs, coils, and even galaxies.

Mathematical curves are defined by equations that express a relationship between at least two variables. In the case of the Spiral of Archimedes, the equation \(r=k\theta\) forms a spiral because as the angle \(\theta\) increases, the radius \(r\) grows linearly with \(\theta\). Its simplistic beauty arises from this direct proportionality between the variables.

Understanding the nature of a curve through its equation is fundamental in mathematics. When you grasp the interplay between the parameters of an equation, you unlock a deeper appreciation of the curve's form and the underlying principles it represents.