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

The spiral unwinds in the counterclockwise direction for \(c>0\).

Step 1: Understanding the Polar Equation

The given equation is in polar coordinates: \( r = \frac{c}{\theta} \). This means as \( \theta \) changes, the distance \( r \) from the origin also changes. We need to sketch this curve to understand its behavior.

Step 2: Analyzing the Behavior of the Spiral

Consider the behavior of \( r \) as \( \theta \) approaches 0. As \( \theta \to 0^+ \), \( r \to \infty \), which suggests that the spiral starts near the pole (origin) and unwinds as \( \theta \) increases.

Step 3: Observing Directional Changes

To determine the direction in which the spiral unwinds, consider how \( r \) changes for increasing \( \theta \). If \( c>0 \), increasing \( \theta \) means \( r \) decreases smoothly from \( \infty \) toward the origin, tracing a path that begins near the horizontal axis and moves counterclockwise.

Step 4: Testing Specific Angles

Try specific values, like \( \theta = \frac{\pi}{2} \), \( \theta = \pi \), and \( \theta = \frac{3\pi}{2} \), to get points on the graph. For instance, at \( \theta = \frac{\pi}{2} \), \( r = \frac{c}{\frac{\pi}{2}} \), placing it in the first quadrant.

Step 5: Drawing the Diagram

Plot the key points determined and draw a smooth curve from the origin outwards, becoming less steep as \( \theta \) increases. The direction of the curve unwinding will be counterclockwise, confirming its path.

Key Concepts

Reciprocal Spiral

The reciprocal spiral is a fascinating curve that can be best understood through its polar equation form, which is given by \( r = \frac{c}{\theta} \). Unlike the familiar Archimedean spiral, where the arm spacing is constant, the reciprocal spiral's spacing decreases as it moves away from the origin. This equation forms the basis of its unique shape and behavior.

In this polar equation, \( r \) represents the radial distance from the origin, and \( \theta \) represents the angle with respect to the positive x-axis. As \( \theta \) increases, \( r \) continues to vary, rendering an intricate path which represents the spiral structure. This characteristic where \( r \) approaches infinity as \( \theta \) approaches zero, creates a unique scenario where the spiral seemingly begins at the pole and extends outward indefinitely.

Direction of Unwinding

The direction of unwinding of the reciprocal spiral is an essential aspect of understanding how the spiral behaves as \( \theta \) changes. For a positive constant \( c \), the reciprocal spiral unwinds in a counterclockwise direction as \( \theta \) increases. This occurs because as \( \theta \) grows, \( r = \frac{c}{\theta} \) decreases.

To visualize this, think of the angle \( \theta \) as the hand of a clock moving backwards, contrary to time, causing the spiral arms to trace outwards from the pole. The reciprocal spiral is unique in its path and illustrates the diverse applications of polar coordinate systems beyond simple circles and ellipses. Consequently, the direction of unwinding is opposite to a more common clockwise orientation seen in other spirals.

Polar Equation

Polar equations offer a different perspective compared to Cartesian coordinates, emphasizing the relationship between radius and angle from a fixed point, the pole. In the equation \( r = \frac{c}{\theta} \), \( c \) is a constant, and it influences how tight or loose the spiral appears.

This type of equation provides an effective means to describe curves that are symmetrical and rotational. Unlike Cartesian equations where x and y are used, polar equations provide a direct relationship between radial distance and angular position. Consequently, polar coordinates are particularly well-suited for natural phenomena and engineering applications, where relations between distance and angle are key.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points using radial distance and angles, rather than traditional x and y coordinates. For the reciprocal spiral \( r = \frac{c}{\theta} \), graphing begins at the pole; as \( \theta \) approaches zero, \( r \) becomes infinitely large, creating a starting point that's close to the origin.

To properly sketch this spiral, it’s beneficial to compute values of \( r \) at various \( \theta \) intervals, such as \( \theta = \frac{\pi}{4} \) or \( \theta = \frac{\pi}{2} \). These calculated points guide the unending path of the spiral, helping to plot the curve accurately. As you connect these points, the continuous and smooth path of the curve emerges, showcasing the spiral's characteristic loops and gradually diminishing arm spacing. This methodical approach of graphing highlights the unique capabilities of polar coordinates to describe complex curves.