Problem 19 - Thomas Calculus

Question

Find parametric equations and a parameter interval for the motion of a particle that starts at \((a, 0)\) and traces the circle \(x^{2}+y^{2}=a^{2}\) a. once clockwise. b. once counterclockwise. c. twice clockwise. d. twice counterclockwise. (There are many ways to do these, so your answers may not be the same as the ones in the back of the book.)

Step-by-Step Solution

Answer

a. \(x = a\cos t; y = -a\sin t; 0 \leq t \leq 2\pi\). b. \(x = a\cos t; y = a\sin t; 0 \leq t \leq 2\pi\). c. \(x = a\cos t; y = -a\sin t; 0 \leq t \leq 4\pi\). d. \(x = a\cos t; y = a\sin t; 0 \leq t \leq 4\pi\).

Step 1: Understanding the Problem

The task requires us to find parametric equations that describe the motion of a particle along a circle defined by \(x^2 + y^2 = a^2\), starting at \((a, 0)\). We also need to define the interval for the parameters to ensure the particle completes the specified number of rotations in either clockwise or counterclockwise direction.

Step 2: Circle Parametrization using Angle

The standard parametrization of a circle \(x^2 + y^2 = a^2\) is \(x = a\cos \theta\) and \(y = a\sin \theta\) where \(\theta\) represents the angle from the positive x-axis. This parametrization moves counterclockwise as \(\theta\) increases.

Step 3: Parametrize One Counterclockwise Rotation

To describe one complete counterclockwise rotation starting at \((a, 0)\), set \(\theta = 0\) to \(\theta = 2\pi\). Therefore, the parametric equations are \(x = a\cos t\) and \(y = a\sin t\) with the parameter interval \(0 \leq t \leq 2\pi\).

Step 4: Parametrize One Clockwise Rotation

For one clockwise rotation, reverse the direction by using \(\theta = -t\). Thus, \(x = a\cos(-t) = a\cos t\) and \(y = a\sin(-t) = -a\sin t\). The parameter interval is \(0 \leq t \leq 2\pi\).

Step 5: Parametrize Two Clockwise Rotations

To complete two clockwise rotations starting at \((a, 0)\), continue with the equation \(x = a\cos t\) and \(y = -a\sin t\) but extend the parameter interval to \(0 \leq t \leq 4\pi\).

Step 6: Parametrize Two Counterclockwise Rotations

To perform two counterclockwise rotations, use \(x = a\cos t\) and \(y = a\sin t\) with an extended parameter interval \(0 \leq t \leq 4\pi\).

Key Concepts

Circle Parametrization

Parametrizing a circle involves using trigonometric functions to describe the position of a point moving along the path of the circle. For a circle centered at the origin with a radius of \(a\), the parametric equations are given by \(x = a\cos \theta\) and \(y = a\sin \theta\). These equations use the angle \(\theta\) to determine the position at any point in the circular path.
This parametrization is based on the Pythagorean identity, where \(\cos^2 \theta + \sin^2 \theta = 1\), ensuring that the equation \(x^2 + y^2 = a^2\) of the circle holds true at all times.
By setting \(\theta\) to vary over different intervals, we can control how the particle moves along the circle, either completing full rotations or partial ones.

Clockwise and Counterclockwise Motion

When dealing with circular motion, the direction the particle takes around the circle is crucial. **Counterclockwise motion** is the natural motion when \(\theta\) increases from 0, moving the particle upwards, around to the left, and down, in a smooth curve. In parametric terms, this uses \(x = a\cos t\) and \(y = a\sin t\), ensuring a standard counterclockwise path as \(t\) increases.
For **clockwise motion**, the particle moves in the opposite direction. This involves negating the angle in the sine component, leading to \(x = a\cos t\) and \(y = -a\sin t\). Here, \(y = -a\sin t\) flips the sine wave, causing the path to move downwards before looping back counterclockwise.

Parameter Interval

To ensure the particle completes its intended path on the circle, we must set an appropriate interval for the parameter \(t\). The interval determines how many times and in which direction the circuit is completed.
For **one full rotation**, whether clockwise or counterclockwise, \(t\) should vary from 0 to \(2\pi\). This represents the complete angle around a circle. If the motion is to be executed twice, the parameter \(t\) extends to \(0\) to \(4\pi\), covering the circle twice.

• **Counterclockwise once**: \(0 \leq t \leq 2\pi\)
• **Clockwise once**: \(0 \leq t \leq 2\pi\)
• **Counterclockwise twice**: \(0 \leq t \leq 4\pi\)
• **Clockwise twice**: \(0 \leq t \leq 4\pi\)

These parameter intervals are essential for correctly time-mapping the motion for applications or understanding.

Trigonometric Functions

The core of circle parametrization is the use of trigonometric functions, specifically sine and cosine. These functions help transform linear angle measures into circular motion patterns, connecting angles with the geometric representation of circles.
**Cosine function** \(\cos \theta\) maps the x-axis, e.g., moving in and out from 1 through to -1. It represents horizontal displacement in the unit circle.
**Sine function** \(\sin \theta\) maps the y-axis, similar behavior with peaks and troughs between 1 and -1, but is noted for its vertical displacement in circular motion.
The cyclical nature of these functions makes them perfect for modeling repeated circular motions, such as those described in parametric equations, thus ensuring smooth, natural motion all around the circle.