Problem 25 - Algebra and Trigonometry

Question

\(25-28\) . The position of an object in circular motion is modeled by the given parametric equations. Describe the path of the object by stating the radius of the circle, the position at time \(t=0\) , the orientation of the motion (clockwise or counterclockwise), and the time \(t\) that it takes to complete one revolution around the circle. $$ x=3 \cos t, \quad y=3 \sin t $$

Step-by-Step Solution

Answer

The path is a circle with radius 3, starting at (3, 0), moving counterclockwise, completing a revolution in \( 2\pi \) time units.

Step 1: Identify the Parametric Equations

The given parametric equations are \( x = 3 \cos t \) and \( y = 3 \sin t \). These equations describe a set of coordinates on a plane as a function of the variable \( t \). This setup is typical for circular motion.

Step 2: Determine the Radius

The equations are of the form \( x = a \cos t \) and \( y = a \sin t \), which represent a circle with radius \( a \). In this case, \( a = 3 \). So, the radius of the circle is 3 units.

Step 3: Find the Position at \( t=0 \)

Substitute \( t=0 \) into the equations: \( x = 3 \cos 0 = 3 \times 1 = 3 \) and \( y = 3 \sin 0 = 3 \times 0 = 0 \). Thus, the starting position at \( t=0 \) is the point (3, 0).

Step 4: Determine the Orientation of Motion

For \( x = 3 \cos t \) and \( y = 3 \sin t \), as \( t \) increases, the point moves from (3, 0) around the circle. This is a counterclockwise movement since the parameterization follows the standard trigonometric orientation.

Step 5: Calculate the Time for One Revolution

The time to complete one full revolution is determined by the period of the sine and cosine functions. Since these functions have a period of \( 2\pi \), the object takes \( 2\pi \) units of time to complete one revolution.

Key Concepts

Radius of a Circle

The radius of a circle is a crucial element in understanding circular motion. In parametric equations like \( x = 3 \cos t \) and \( y = 3 \sin t \), the radius can be quickly identified. Here, the radius is the value 3, which is the coefficient of both \( \cos t \) and \( \sin t \). This tells us that every point on the path described by these equations is 3 units away from the center of the circle.

• The formula used is generally \( x = a \cos t \) and \( y = a \sin t \), where \( a \) represents the radius.
• The radius indicates the size of the circle: a larger radius means a larger circle.

A clear understanding of the radius helps us visualize how expansive or condensed the circular motion will be.

Orientation of Motion

Orientation of motion refers to the direction in which an object moves along its circular path. For the given equations \( x = 3 \cos t \) and \( y = 3 \sin t \), this orientation is determined by how \( t \), the parameter, increases. As \( t \) progresses from 0 to \( 2\pi \), the object moves counterclockwise.

• Counterclockwise is the typical direction in mathematics as it follows the positive change of angles.
• Clockwise movement occurs when the parameterization is modified, often using negative signs or reversing the equations.

Grasping the orientation helps in predicting the path and ultimate position of the object as time increases.

Trigonometric Functions

Trigonometric functions are central to describing circular motion and parametric equations. In the equations \( x = 3 \cos t \) and \( y = 3 \sin t \), \( \cos \) and \( \sin \) encode the smooth, wave-like motion inherent in circles. Here's why they are used:

• They offer periodic behavior, which suits the repeating nature of circles.
• \( \cos \) and \( \sin \) represent horizontal and vertical positions, projecting points onto a circle.
• They help in computing angles and arcs, fundamental to circular paths.

Understanding these functions allows us to connect circular paths to the underlying algebraic and geometrical principles.

Circular Motion

Circular motion is an elegant motion where an object moves around the circumference of a circle. Parametric equations like \( x = 3 \cos t \) and \( y = 3 \sin t \) are a mathematical expression of this motion in two dimensions. Key features:

• The motion is continuous and repeats, forming a circular path.
• Each full loop or revolution returns the object to its starting position.
• The object's path is entirely defined by its radius and the parametric functions.

By learning about circular motion, students can better appreciate how objects move in planetary orbits, wheels turning, and various engineering and physics applications.

Period of Sine and Cosine

The period of a trigonometric function is the interval after which it repeats its values. For both \( \sin t \) and \( \cos t \), this period is \( 2\pi \). In the parametric equations at hand, this period directly influences the time an object takes to complete its revolution around the circle.

• The complete '+one loop' of the circle corresponds to \( t = 0 \) to \( t = 2\pi \).
• The period remains consistent regardless of the amplitude or the coefficient in front of \( \cos \) or \( \sin \).
• Knowing this period helps in predicting the time dynamics of circular motion.

Grasping the period concept is crucial for understanding time-dependent phenomena in physics, such as waves, rotations, and oscillatory motions.