Problem 36 - Calculus Early Transcendentals

Question

Find parametric equations that describe the circular path of the following objects. Assume \((x, y)\) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There is more than one way to describe any circle. A Ferris wheel has a radius of \(20 \mathrm{m}\) and completes a revolution in the clockwise direction at constant speed in 3 min. Assume that \(x\) and \(y\) measure the horizontal and vertical positions of a seat on the Ferris wheel relative to a coordinate system whose origin is at the low point of the wheel. Assume the seat begins moving at the origin.

Step-by-Step Solution

Answer

Answer: The parametric equations for the given Ferris wheel are: \(x = 20 \cos\left(-\frac{2\pi}{3}t\right)\) \(y = 20 \sin\left(-\frac{2\pi}{3}t\right)\) where \(t\) is time in minutes, and (x, y) represents the position of a seat relative to the coordinate system's origin at the low point of the wheel.

Step 1: Convert revolution time to angular speed

Since the Ferris wheel takes 3 minutes to complete a revolution, we can find the angular speed in radians per minute. Remember that one revolution equals \(2\pi\) radians. Angular speed (ω) = \(\frac{2\pi}{\text{revolution time}} = \frac{2\pi}{3}\) radians per minute.

Step 2: Modify the parametric equations for a clockwise direction

As the Ferris wheel is moving in a clockwise direction but the standard parametric equations assume a counterclockwise movement, we can adjust the equations by changing the sign of the angle in the equations: \(x = 20 \cos(-\omega t)\) \(y = 20 \sin(-\omega t)\)

Step 3: Substitute the angular speed into the parametric equations

Now, we can substitute the angular speed (ω) we found in step 1 into the parametric equations: \(x = 20 \cos\left(-\frac{2\pi}{3}t\right)\) \(y = 20 \sin\left(-\frac{2\pi}{3}t\right)\) These are the parametric equations describing the circular path of a seat on the given Ferris wheel. Note that \(t\) represents time in minutes, and \((x, y)\) represents the position of a seat relative to the coordinate system's origin at the low point of the wheel.

Key Concepts

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In this context, the Ferris wheel seat travels around the circle, maintaining a constant distance, which is the radius of the Ferris wheel, from the center. The path is perfectly circular and any point on a circle can be described by two coordinates, horizontal (x) and vertical (y), relative to an origin.
This type of motion has unique features:

• Constant Speed: The seat completes each circle in the same amount of time, which means its speed is constant.
• Periodic Movement: After a certain period, here 3 minutes, the motion repeats itself as the seat returns to its starting position.
• Direction of Rotation: In this problem, the Ferris wheel rotates clockwise. This affects how we calculate the parametric equations using negative angles.

Understanding these features helps in comprehending how objects like the seats on a Ferris wheel maintain their periodic paths.

Angular Speed

Angular speed is a measure of how fast an object is moving through an angle, typically measured in radians per unit of time. It's crucial to relate circular motion in terms of angles rather than traditional linear velocity. For the Ferris wheel, we calculated the angular speed as follows:

• One complete revolution is equivalent to an angle of \(2\pi\) radians.
• If the Ferris wheel completes one revolution in 3 minutes, the angular speed \(\omega\) is \(\frac{2\pi}{3}\) radians per minute.

The angular speed informs how quickly the Ferris wheel rotates, and it’s a key value used to write the parametric equations for the motion. With angular speed, we can transform the continuous spinning motion into algebraic expressions that describe the seat's location at any moment.

Coordinate System

Coordinate systems allow us to specify locations in space. For this problem, we use a two-dimensional coordinate plane to track the Ferris wheel’s seat position. Here's how it works:

• Origin: The origin is set at the ferris wheel's lowest point. All positions \((x, y)\) are measured from here.
• Horizontal Axis (x): Represents how far left or right the seat is from the origin.
• Vertical Axis (y): Represents how high or low the seat is from the origin.

Using a coordinate system simplifies finding exact positions, as it allows us to calculate both \(x\) and \(y\) using trigonometric functions. By integrating angular speed, we pinpoint how these positions change with time, tracing out the circular path. Adjusting the equations for clockwise movement requires subtly shifting our calculations, as was done in the solution with the negative angle adjustments.