Problem 79 - 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. \( (x, y) = (a \cos t, -a \sin t), \ t=[0, 2\pi] \) b. \( (x, y) = (a \cos t, a \sin t), \ t=[0, 2\pi] \) c. \( (x, y) = (a \cos t, -a \sin t), \ t=[0, 4\pi] \) d. \( (x, y) = (a \cos t, a \sin t), \ t=[0, 4\pi] \)

Step 1: Understand the Circle's Equation

The equation \( x^2 + y^2 = a^2 \) represents a circle centered at the origin with a radius of \( a \). The parametric equations for a circle can be given as \( x = a \cos t \) and \( y = a \sin t \), where \( t \) is a parameter, usually the angle in radians.

Step 2: Parametric Equations for Clockwise Motion

For clockwise motion, we need the angle to decrease as the parameter \( t \) increases. Thus, we use \( x = a \cos t \) and \( y = -a \sin t \). To start at \( (a, 0) \), we use \( t = 0 \) as the starting point.

Step 3: Parameter Interval for Once Clockwise

To trace the circle once in a clockwise direction, the parameter \( t \) should go from \( 0 \) to \( 2\pi \). This makes a complete clockwise rotation.

Step 4: Parametric Equations for Counterclockwise Motion

For counterclockwise motion, the angle increases as \( t \) increases. The parametric equations become \( x = a \cos t \) and \( y = a \sin t \), reflecting the natural coordinate system.

Step 5: Parameter Interval for Once Counterclockwise

To travel once counterclockwise, \( t \) should go from \( 0 \) to \( 2\pi \). This provides a full rotation in the counterclockwise direction.

Step 6: Twice Clockwise Movement

To go twice around the circle in a clockwise motion, \( t \) should range from \( 0 \) to \( 4\pi \). This completes two full clockwise rotations.

Step 7: Twice Counterclockwise Movement

For two counterclockwise rotations, the parameter \( t \) needs to go from \( 0 \) to \( 4\pi \), resulting in two full counterclockwise circuits.

Key Concepts

Circle Equation

A circle equation in the Cartesian coordinate system can be simply expressed as \(x^2 + y^2 = a^2\), where \(a\) is the radius of the circle. This equation describes a perfect circle centered at the origin \((0,0)\). The values of \(x\) and \(y\) are the coordinates of any point lying on the circle. When dealing with circles in parametric form, we shift to using trigonometric functions because they readily describe circular motion.

To represent the circle parametrically, we use:

• \(x = a \cos t\)
• \(y = a \sin t\)

Here, \(t\) acts as the parameter and usually represents the angle in radians relative to the positive \(x\)-axis. As \(t\) ranges from 0 to \(2\pi\), the parametric equations trace out the entire circle once. Parametric equations offer a versatile way to describe the motion along any path, especially circular ones.

Clockwise Motion

Clockwise motion, as the name suggests, is movement that mimics the direction of clock hands. To achieve this in parametric equations for a circle, we must ensure that the angle decreases as we increase the parameter \(t\).

For clockwise motion starting from \((a, 0)\), the equations are:

• \(x = a \cos t\)
• \(y = -a \sin t\)

This slight adjustment in the sign of the sine function allows the particle to trace the circle in a clockwise direction.
To complete one rotation clockwise, we set \(t\) to range from 0 to \(2\pi\). This interval effectively makes a complete loop in a clockwise manner.

Counterclockwise Motion

Counterclockwise motion naturally aligns with the typical coordinate system orientation. In mathematics, angles usually increase in the counterclockwise direction, making it more straightforward to describe.

The standard parametric equations for counterclockwise movement from \((a, 0)\) are:

• \(x = a \cos t\)
• \(y = a \sin t\)

This usage follows the natural circular trajectory. Unlike clockwise motion, the signs remain unchanged for both components.
For a full counterclockwise rotation, \(t\) should also range from 0 to \(2\pi\), completing one entire loop along the circle.

Complete Rotation

A complete rotation on a circular path means returning to the starting point after tracing the circle fully. This is characterized by the parameter \(t\) covering the interval from 0 to \(2\pi\), whether it's clockwise or counterclockwise.

If you want multiple rotations, such as twice around the circle:

• For clockwise, \(t\) spans from 0 to \(4\pi\)
• For counterclockwise, the same, \(t\) ranges from 0 to \(4\pi\)

Each full \(2\pi\) span accounts for one complete round. By extending the interval, you can dictate how many rotations the path involves. Understanding this is crucial for defining the particle's trajectory within any circular path.