Problem 27 - 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. $$x=\sin 2 t, \quad y=\cos 2 t$$

Step-by-Step Solution

Answer

The circle has radius 1, initial position \((0,1)\), moves counterclockwise, and takes \(t=\pi\) for one revolution.

Step 1: Identify the Parametric Equations

The given parametric equations are \(x = \sin 2t\) and \(y = \cos 2t\). These equations describe the motion of the object in terms of \(x\) and \(y\) coordinates as functions of time \(t\).

Step 2: Express as a Circular Equation

To find the equation of the path, we use the trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\). Thus, substitute \(x = \sin 2t\) and \(y = \cos 2t\) into this identity: \(x^2 + y^2 = \sin^2 2t + \cos^2 2t = 1\). This shows the path is a circle with equation \(x^2 + y^2 = 1\).

Step 3: Determine the Radius of the Circle

The general form of a circle is \(x^2 + y^2 = r^2\). Comparing it with our equation \(x^2 + y^2 = 1\), we find the radius \(r = 1\).

Step 4: Find the Initial Position at \(t=0\)

Substitute \(t = 0\) into the parametric equations: \(x = \sin(0) = 0\) and \(y = \cos(0) = 1\). Thus, the initial position is \((0, 1)\).

Step 5: Determine the Orientation of the Motion

To find if the motion is clockwise or counterclockwise, check the derivatives: the derivative \(dx/dt = 2\cos 2t\) and \(dy/dt = -2\sin 2t\). At \(t = 0\), \(dx/dt = 2\) and \(dy/dt = 0\), indicating motion towards positive \(x\), which is counterclockwise around the circle.

Step 6: Calculate the Time for One Revolution

Since the parameter \(2t\) completes a cycle from \(0\) to \(2\pi\), it takes \(2t = 2\pi\) for a full revolution. Therefore, \(t = \pi\) for one complete revolution.

Key Concepts

Circular Motion

Objects in circular motion follow a path that is defined by a constant distance from a fixed point, known as the center of the circle. This distance is referred to as the radius of the circle. In our case, the parametric equations given were

• \( x = \sin 2t \)
• \( y = \cos 2t \)

These indicate that the object moves in a circle. By utilizing the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), it becomes evident that the equation \( x^2 + y^2 = 1 \) is the resulting path, confirming a circle with a radius of 1 unit.
The initial position at time \( t = 0 \) is determined by substituting the value into the parametric equations, ensuring we start tracking the motion from a specific known point, here

• Position at \( t = 0 \) is \( (0, 1) \)

Understanding these properties allows for visualizing how the object traverses its circular path, continuously moving around the center.

Trigonometric Identity

Trigonometric identities serve as powerful tools to simplify and interpret functions that are modeled using trigonometric terms. One key identity applied in the context of circular motion is

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity ensures that for any angle \( \theta \), the sum of the squares of the sine and cosine is always 1.
In parametric equations representing circular motion, such as \( x = \sin 2t \) and \( y = \cos 2t \), replacing \( x \) and \( y \) in the trigonometric identity results in a standard form of a circle: \( x^2 + y^2 = 1 \).
This signifies the path is indeed circular, with a radius derived from comparing it to the general circle equation \( x^2 + y^2 = r^2 \). Adopting trigonometric identities not only provides proofs but also simplifies the equations to easily discern key characteristics of motion, like shape and size of its path.

Derivatives

In the study of parametric equations, derivatives give important information about the rate of change of the coordinates with respect to time, and thus about the path's orientation and dynamics.
The derivatives of the coordinates \( x \) and \( y \) with respect to time are

• \( \frac{dx}{dt} = 2\cos 2t \)
• \( \frac{dy}{dt} = -2\sin 2t \)

These derivatives help determine the direction of motion. Observation at \( t = 0 \) reveals

• \( \frac{dx}{dt} = 2 \)
• \( \frac{dy}{dt} = 0 \)

indicating that the motion is initially directed towards the positive \( x \)-axis, confirming a counterclockwise path.
Furthermore, understanding how parameter \( 2t \) affects time enables calculation of the period, or time taken for a complete revolution around the circle. Here, \( 2t = 2\pi \) leads to \( t = \pi \), denoting the time for one full cycle.