Problem 28 - 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. $$ x=4 \cos 3 t, \quad y=4 \sin 3 t $$

Step-by-Step Solution

Answer

Radius: 4, Position: (4, 0), Motion: Counterclockwise, Time: \(\frac{2\pi}{3}\).

Step 1: Identify the Circle's Radius

The parametric equations given are \(x = 4 \cos 3t\) and \(y = 4 \sin 3t\). These are of the form \(x = r \cos(\omega t)\) and \(y = r \sin(\omega t)\), where \(r\) is the radius. Here, \(r = 4\). So, the radius of the circle is 4.

Step 2: Determine the Position at Time t=0

To find the position at \(t=0\), substitute \(t=0\) into the parametric equations: \(x(0) = 4\cos 0 = 4\) and \(y(0) = 4\sin 0 = 0\). So, the position is at \((4, 0)\).

Step 3: Establish the Motion's Orientation

The sign of the coefficient of \(t\) in \(\omega t\) indicates the motion's orientation. In this case, since \(\omega t = 3t\) and is positive, the motion is counterclockwise.

Step 4: Calculate Time for One Revolution

The coefficient of \(t\) in the argument of the trigonometric functions is \(3\), which is the angular speed \(\omega\). The time for one complete revolution is \(\frac{2\pi}{\omega}\). So, substituting \(\omega = 3\), we get \(T = \frac{2\pi}{3}\).

Key Concepts

Radius of the Circle

In the realm of circular motion with parametric equations, understanding the radius is key. The radius is a fixed distance from the center to any point on the circle. In parametric equations like the ones given, the radius can be directly determined. The equations are formatted as:

• \( x = r \cos(\omega t) \)
• \( y = r \sin(\omega t) \)

In our example, those equations are \( x = 4 \cos(3t) \) and \( y = 4 \sin(3t) \). The constant "4" in front of the trigonometric functions represents the circle's radius. Therefore, for this circle, the radius \( r \) is 4 units. This defines how far any point on the circle is from the center.

Orientation of Motion

The orientation of motion tells us which direction the object moves along its circular path. In the case of parametric equations, the term \( \omega t \) gives away the secret of the path's orientation. If the coefficient of \( t \), denoted as \( \omega \), is positive, then the motion is counterclockwise. Conversely, if it is negative, the motion is clockwise.

For the given equations \(x = 4 \cos(3t)\) and \(y = 4 \sin(3t)\), the coefficient of \( t \) is 3, which is positive. Therefore, the orientation of the motion is counterclockwise. Visualizing this helps in knowing the rotational direction around the circle.

Parametric Equations

Parametric equations provide a robust way to describe motion in two dimensions, especially in circular paths. They express both the x and y coordinates as functions of a third variable, usually time \( t \). This gives a dynamic picture of how an object moves.

For instance, in the equations \(x = 4 \cos(3t)\) and \(y = 4 \sin(3t)\), time \( t \) creates a relationship between the horizontal and vertical positions. As \( t \) varies, we can trace the object's path on the circle. This combination of cosine and sine functions beautifully illustrates motion as it cycles through angles, completing a full revolution around the circle.

Angular Speed

Angular speed, denoted as \( \omega \), is a key player in parametric equations for circular motion. It measures how fast the object travels along its circular path. It specifically refers to how many radians the object sweeps per unit time. In the parametric equations format \( x = r \cos(\omega t) \) and \( y = r \sin(\omega t) \), \( \omega \) is the coefficient of \( t \).

In the problem at hand, \( \omega = 3 \). To find the time it takes for one full revolution, we calculate the period \( T \) as \( T = \frac{2\pi}{\omega} \). Substituting \( \omega = 3 \), we have \( T = \frac{2\pi}{3} \). This elongates the beauty of trigonometry and physics where circular motion converges, demonstrating how quickly the circle is completed.