Problem 49 - Contemporary Precalculus

Question

The original Ferris wheel, built by George Ferris for the Columbian Exposition of \(1893,\) was much larger and slower than its modern counterparts: It had a diameter of 250 feet and contained 36 cars, each of which held 60 people; it made one revolution every 10 minutes. Imagine that the Ferris wheel revolves counterclockwise in the \(x-y\) plane with its center at the origin. A car had coordinates (125,0) at time \(t=0 .\) Find the rule of a function that gives the \(y\) -coordinate of the car at time \(t.\)

Step-by-Step Solution

Answer

Answer: The function representing the y-coordinate of the car at any time t is given by: $$ y(t)=125\ \sin\left(\frac{2\pi}{600}t\right) $$

Step 1: Calculate the angular speed of the car

Since the car takes 10 minutes for a full revolution, we will convert it to seconds for simplification. So, the car takes \(10\times60=600\) seconds to complete a full circle. The angular speed \(\omega\) is equal to the angle made by the car at 1 second. In a full circle, there are \(360^\circ\) or \(2\pi\) radians. Therefore, the angular speed \(\omega\) in radians per second can be calculated as follows: $$ \omega = \frac{2\pi}{600} $$

Step 2: Find the angle of rotation at time \(t\)

At time \(t\), the angle of rotation with respect to the initial position can be found by multiplying the angular speed with time \(t\). Let's denote this angle by \(\theta\): $$ \theta = \omega t = \frac{2\pi}{600}t $$

Step 3: Use trigonometry to find the \(y\)-coordinate

Since the car moves in a circular path, we can use trigonometric functions to find the \(y\)-coordinate. In this case, we can use the sine function because it represents the \(y\)-coordinate in a unit circle. The radius of the Ferris wheel is 125 feet. So, the \(y\)-coordinate of the car at any given time \(t\) can be represented as: $$ y(t)=125\ \sin(\theta) = 125\ \sin\left(\frac{2\pi}{600}t\right) $$ Now we have the function that describes the \(y\)-coordinate of the car at any time \(t\): $$ y(t)=125\ \sin\left(\frac{2\pi}{600}t\right) $$

Key Concepts

Angular Speed

Angular speed is an essential concept when dealing with objects in circular motion, like a Ferris wheel. It describes how quickly an object moves through an angle. In simple terms, angular speed tells us how fast the car on the Ferris wheel is rotating in terms of degrees or radians per unit of time. For this exercise, the formula for angular speed is

• \(\omega = \frac{2\pi}{T}\)

where \(T\) is the time taken for one complete revolution. Since the Ferris wheel takes 10 minutes per revolution or 600 seconds, the angular speed \(\omega\) is \(\frac{2\pi}{600}\) radians per second. Breaking it down into

• seconds makes calculations more straightforward,
• and using radians is standard for mathematical purposes.

Ferris Wheel Mathematics

Ferris wheel mathematics involves applying mathematical principles to understand the motion and behavior of Ferris wheels. At the core of this application are concepts like angular speed, radius, and motion equations. The Ferris wheel’s diameter is 250 feet, meaning the radius is half of that, which is 125 feet. With the center of the Ferris wheel at the origin, we can say:

• Every point on this structure is coordinate-based.
• The farthest points extend to \((\pm125, 0)\) and \((0, \pm125)\).
• Its motion, whether clockwise or counterclockwise, is described through mathematical equations that use trigonometry.

The initial position of the car at \((125, 0)\) helps define this motion over time. Using trigonometric functions showcases how far above or below the center of the wheel the car is located at any specific time \(t\).

Circular Motion

Circular motion describes the movement of an object along the circumference of a circle or a circular path. This movement is crucial when understanding objects moving in a circle, like cars on Ferris wheels. Key elements of circular motion include:

• Radius: In our case, the radius of the Ferris wheel is 125 feet.
• Angular Speed: Determines how fast the object is rotating around the circle.
• Path: It is circular and the car remains at the same distance from the center as it moves.
• Time: It affects the position of the car as it revolves; one complete revolution takes the car from its start point and back.

Making one full path implies returning to the same angle but carrying the dynamic position the entire time you're on this circular route.

Sine Function

The sine function is crucial in determining the vertical position of objects moving in circular paths. When applying trigonometry to circular motion, the sine function helps find the car's \(y\)-coordinate on the Ferris wheel. The equation is \[y(t) = 125\sin(\theta)\] where \(\theta\) is the angle in radians at any time \(t\). The sine function draws on the principle of a unit circle where:

• \(\sin(\theta) = \frac{opposite}{hypotenuse}\)
• \(\theta\) changes as time progresses, translating to new positions.

The outcome is that as the wheel revolves, the car's position regarding height above the ground varies woodenly, hitting maximum and minimum values, determined by the wheel's radius. Utilizing this info, we can model not only the Ferris wheel's rotational dynamics but the periodic ups and downs that a passerby might feel.