Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the reciprocal spiral given by \(r=c / \theta\). For \(c>0\), does it unwind in the clockwise direction?

Step-by-Step Solution

Answer

Yes, it unwinds in the clockwise direction for \(c > 0\).

Step 1: Understanding the Polar Equation

The given polar equation is \( r = \frac{c}{\theta} \). This equation describes a reciprocal spiral, where \( r \) is the radial distance and \( \theta \) is the angle. The constant \( c \) is positive.

Step 2: Rewriting Polar to Cartesian Coordinates

To sketch the graph, we can rewrite the polar equation in terms of Cartesian coordinates. The relationships are \( x = r \cos \theta \) and \( y = r \sin \theta \). First, express \( r \) as \( r = \frac{c}{\theta} \), and then substitute \( r \) in the equations for \( x \) and \( y \).

Step 3: Substitution and Plotting

Substitute \( r \) into the Cartesian coordinates: \( x = \frac{c}{\theta} \cos \theta \) and \( y = \frac{c}{\theta} \sin \theta \). As \( \theta \) ranges from positive to negative infinity, calculate corresponding \( x \) and \( y \) values to observe the path of the spiral.

Step 4: Analyzing the Direction of the Spiral

As \( \theta \) increases from 0 to positive infinity, \( r \) decreases, causing the spiral to wind closer to the origin. As \( \theta \) moves from negative to positive infinity, the angle increases, but the distance \( r \) decreases, indicating the spiral unwinds in the clockwise direction for \( c > 0 \).

Step 5: Graphically Sketch the Reciprocal Spiral

Plot several points using calculated values from substitution, and connect these points to form the spiral shape. Notice that the spiral does unwind clockwise as \( \theta \) increases, confirming the direction due to the relationship \( r = \frac{c}{\theta} \) and \( c > 0 \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of recording the position of a point in a plane, using a distance and an angle. They are different from the more commonly used Cartesian coordinates, which utilize perpendicular x and y axes. In polar coordinates, a point is described by how far it is from the origin (denoted as \( r \)) and the angle (\( \theta \)) formed with the positive x-axis.
This system is particularly useful for curves, like spirals, because it emphasizes the radius and angular motion, making patterns more apparent. For instance, the reciprocal spiral specific to our problem has polar coordinates defined by the equation \( r = \frac{c}{\theta} \). This shows how the distance \( r \) from the origin changes inversely with the angle \( \theta \).
This symmetry creates spirals and other circular patterns that naturally find application in fields as diverse as physics, engineering, and even art.

Cartesian Coordinates

Cartesian coordinates provide another way to describe the position of points on a plane. This system uses two numbers: \( x \) and \( y \). The \( x \)-coordinate represents horizontal distance, while the \( y \)-coordinate represents vertical distance.
To convert polar coordinates to Cartesian coordinates, we use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

By applying these, we transform the polar equation \( r = \frac{c}{\theta} \) into Cartesian coordinates: \( x = \frac{c}{\theta} \cos \theta \) and \( y = \frac{c}{\theta} \sin \theta \).
This conversion facilitates graphing the spiral on a standard x-y plane, which is familiar to most people.

Graphing Polar Equations

Graphing polar equations involves plotting points based on their radial distance and angle. To graph the reciprocal spiral, follow these steps:

• Choose several values of \( \theta \), including both positive and negative.
• Calculate corresponding \( r \) values using \( r = \frac{c}{\theta} \).
• Convert each calculated pair of \( (r, \theta) \) into Cartesian coordinates \( (x, y) \) using \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Plot each \( (x, y) \) point on a Cartesian plane.
• Connect the points smoothly to reveal the spiral pattern.

As you graph, you’ll see how the spiral loops closer or further away from the origin. This method of graphing makes it tangible to analyze how changes in \( \theta \) and other factors alter the shape.

Direction of Spiral Winding

The direction of spiral winding refers to how the spiral behaves as \( \theta \) varies. In our case, we're dealing with a reciprocal spiral defined by \( r = \frac{c}{\theta} \). As \( \theta \) grows larger from a small value, \( r \) shrinks, and the spiral moves closer to the origin and winds clockwise.
Similarly, as \( \theta \) changes from negative to positive, the spiral continues this inward-closing motion. For positive \( c \), this results in a clockwise unwinding. It's a fascinating effect, as it demonstrates the interplay between radial distance and angular position.
Understanding the direction of winding is essential in predicting and utilizing the behavior of spirals in complex systems and designs.