Problem 102 - Precalculus : Building Concepts and Connections

Question

The first ferris wheel was 250 feet in diameter. It was invented by John Ferris in \(1893 .\) Assuming it made one revolution every 30 seconds, what was the angular speed of a passenger (assume the passenger is on the edge of the wheel) in degrees per minute? What was the passenger's linear speed in feet per minute?

Step-by-Step Solution

Answer

The passenger's angular speed was \( \frac{2\pi}{30}*57.296*60 \) degrees per minute, and his linear speed was \( 125*2\pi/30*60 \) feet per minute.

Step 1: Calculate Angular Speed

First calculate the angular speed. A full revolution is equivalent to \( 2\pi \) radians or 360 degrees. Since the ferris wheel makes a revolution every 30 seconds, its angular speed in radians per second is \( \omega = \theta/t = 2\pi/30 \). However, we are asked to find the angular speed in degrees per minute, so we would need to convert radians per second to degrees per minute. One radian is approximately 57.296 degrees and one minute is 60 seconds, so the final angular speed is \(\frac{2\pi}{30}*57.296*60 \) degrees per minute.

Step 2: Calculate Linear Speed

The formula for linear speed is \(v = r\omega\), where \( r \) is the radius of the ferris wheel and \( \omega \) is its angular speed in radians per second. The radius of the ferris wheel is half of its diameter, which is \( r = 250/2 \) feet. Thus the linear speed in feet per second is \( v = r\omega = 125*2\pi/30 \) feet per second. To get the linear speed in feet per minute, multiply the result by 60. So the final linear speed is \( 125*2\pi/30*60 \) feet per minute.

Key Concepts

Linear Speed

Linear Speed refers to the distance traveled by an object in a specific direction per unit of time. Imagine you are watching a ferris wheel at an amusement park. The linear speed of a passenger on the edge is the speed at which they move along their path.
The formula for linear speed, denoted as \( v \), is \( v = r\omega \), where \( r \) is the radius of the circle and \( \omega \) is the angular speed in radians per second. For the ferris wheel example given, we start by finding the radius as half of its diameter.
Since the diameter of the ferris wheel is 250 feet, its radius \( r \) is 125 feet. Once the angular speed is calculated, multiplying it by the radius gives the linear speed in feet per second. Finally, converting this speed to feet per minute by multiplying by 60 seconds gives us the motion in terms we regularly use.

Radians to Degrees Conversion

Converting radians to degrees is essential, especially when dealing with angular measurements in everyday life, as degrees are more intuitive for most people.
Radians are often used in mathematics as they make calculations easier. A full circle is \( 2\pi \) radians, or 360 degrees. Thus, one radian is approximately 57.296 degrees. To convert from radians to degrees, you can multiply by this factor.
For example, converting the angular speed of a ferris wheel from radians per second to degrees per minute entails a two-step conversion: first, from radians to degrees, and then from seconds to minutes by multiplying by 60 seconds. This double conversion allows us to understand and work with angular speeds in more familiar terms.

Circular Motion

Circular motion involves an object moving along the circumference of a circle. Understanding it helps decipher real-world problems such as roller coasters or ferris wheels.
Key terms include:

• **Centripetal Force**: the force directed towards the center keeping an object in circular path.
• **Angular Speed**: how fast an angle changes over time.
• **Linear Speed**: how fast an object moves over a distance.

For circular motion, both linear and angular speeds are interconnected. Angular speed tells us how many radians or degrees an object covers per unit of time, while linear speed shows the path traveled over the same period. Considering the Ferris wheel's 30-second revolution, we can calculate these values to better analyze motion.

Ferris Wheel Mathematics

Ferris Wheel Mathematics involves using principles of circular motion to understand the dynamics of ferris wheels, an amusement park staple.
Calculations often involve determining angular and linear speeds, radius and diameter, and applying them to real-world scenarios. The first ferris wheel, invented by John Ferris, provides a perfect context for this application.
By rotating every 30 seconds and with a diameter of 250 feet, we derive mathematical insights student can relate to, such as determining how fast a passenger moves in feet per minute, or understanding speed in degrees per minute. Such exercises solidify mathematical concepts while connecting them to tangible, real-life experiences.