Problem 28 - Precalculus Mathematics for Calculus

Question

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

The circle has a radius 4. Initial position is (4, 0). The motion is counterclockwise, taking \(\frac{2\pi}{3}\) to complete a revolution.

Step 1: Identify Parametric Equation Patterns

The given parametric equations are similar to the standard circle equations, where \(x = r \cos \theta\) and \(y = r \sin \theta\). Here, both equations include a coefficient of 4 in front of the trigonometric functions, indicating the radius \(r = 4\).

Step 2: Determine Radius of the Circle

From the standard circle equations, the coefficients of the cosine and sine functions give the radius of the circle. Since both \(x = 4 \cos 3t\) and \(y = 4 \sin 3t\) have the coefficient 4, the radius is \(r = 4\).

Step 3: Determine Initial Position

To find the position at \(t = 0\), substitute \(t = 0\) into both equations. This gives \(x = 4 \cos(3 \times 0) = 4\) and \(y = 4 \sin(3 \times 0) = 0\). Thus, the initial position is \((4, 0)\).

Step 4: Determine Orientation of Motion

To determine the orientation, consider a small value of \(t \gt 0\). Substitute \(t = \frac{\pi}{6}\) as an example. The small positive \(t\) rotates the angle slightly. Calculate \(x = 4 \cos(\frac{\pi}{2}) = 0\) and \(y = 4 \sin(\frac{\pi}{2}) = 4\). This indicates the object is moving in the positive y-direction, confirming counterclockwise motion.

Step 5: Compute Period for One Revolution

To find the time \(t\) it takes to complete one revolution, we need to solve for when the angle in the cosine and sine functions (\(3t\)) completes a full cycle of \(2\pi\), i.e., \(3t = 2\pi\). Solving this gives \(t = \frac{2\pi}{3}\).

Key Concepts

Circular Motion

When we talk about circular motion, we're referring to the motion of an object that travels along a circular path. In this problem, the position of the object is represented with parametric equations. These equations, using trigonometric functions, perfectly describe how the object moves along a circle.

For the given parametric equations, we have:

• The motion is defined by the equations: \(x = 4 \cos 3t\) and \(y = 4 \sin 3t\).
• The trigonometric functions \(\cos\) and \(\sin\) help us determine the x and y coordinates at any given time \(t\).
• The number multiplied before \(t\) (which is 3 in this case) indicates how fast the object moves around the circle.

These equations provide a comprehensive overview of the object's circular path, representing the combination of its x and y positions over time. Understanding these concepts helps in visualizing how the object traverses its path in continuous circular motion.

Radius of a Circle

The radius of a circle is a key concept in determining the size of the circle within which the object moves. The parametric equations \(x = 4 \cos 3t\) and \(y = 4 \sin 3t\) reveal this crucial information.

From these equations:

• The coefficient in front of the cosine and sine functions represents the radius.
• Here, both coefficients are 4, meaning the radius \( r \) is 4.

This constant radius implies that at any point in time, the object is consistently 4 units away from the center of the circle. Knowing the radius is vital for understanding the scale of the object's path and size of the circular motion. It stays unchanged unless the coefficients in the parametric equations change.

Orientation of Motion

The orientation of motion indicates the direction in which an object travels around the circle. This can either be clockwise or counterclockwise. To find this for our object, we assess how the parametric equations change over time.

By substituting a small positive \(t\), like \(t = \frac{\pi}{6}\), in the equations:

• We find \(x = 4 \cos \frac{\pi}{2} = 0\) and \(y = 4 \sin \frac{\pi}{2} = 4\).
• Such calculations indicate movement in the positive y-direction.

Hence, the object is moving counterclockwise. This orientation reveals not only how the object travels but also ensures clarity in the visualization of its consistent path. Orientation in parametric equations is an integral part of interpreting the nature of the object’s movement.

Period of Revolution

The period of revolution describes the duration it takes for the object to complete one full cycle around the circle. For our parametric equations, this period is derived by focusing on the angle within the trigonometric functions.

We have:

• The angle here is \(3t\).
• A full revolution corresponds to \(2\pi\) in radians.
• Solving \(3t = 2\pi\) gives \(t = \frac{2\pi}{3}\).

This result, \(t = \frac{2\pi}{3}\), tells us the precise time needed for completing a circle. Understanding the period of revolution is crucial for predicting when and where the object will be at any point in its cyclical journey. It provides a timeline for the entire motion within the circle.