WEBFEATS II USMA West Point West Point, NY July 2226th, 2002


Bear
Mountain Inn
Bear
Mountain State Park, NY

The
Carousel
Carousels are not exactly considered "thrill machines." Yet, carousels are as reliant on the laws of motion as their more exciting cousins, the roller coasters. Let's take a trip to the Bear Mountain State Park in New York. You are going to take a ride on the all new Bear Mountain Carousel. The carousel was opened in 2001 and is housed in its own private pavilion. After enjoying the ride you should be able to take the data that has been collected and explore the math behind this simplistic amusement park ride.

Linear Motion Problem Kara and Jami took a ride on the Bear Mountain Carousel. Jami selected a horse on the outside row, where Kara chose to sit in the row closest to the center of the carousel. The rows are 17.6 feet and 12.4 feet respectively. The angular speed of the carousel is 4.3 revolutions per minute. in Calculate the linear speed of each rider in feet per second? Who is traveling at a faster rate? Why? (Round answers to the nearest tenth) 
THINK ABOUT IT . . . Are some horses moving faster than others? The carousel is a delicate balance of motion and forces. All of the horses move through one complete revolution in the same amount of time. The horses on the outside of the carousel have to cover more distance than the inside horses in this time. This means the horses on the outside have a faster linear speed than those at the hub. Check out this geometrical representation of the problem 
Equine (that means "horse!") Motion Problem Now that you have analyzed the linear speeds of this carousel you will study the motion as you ride the carousel. However, before you resume your ride on the carousel, refer to the following applet.
Graph
Comparisons
 Compare the graph created in the applet you viewed earlier with your
sketch. How does your sketch differ from the graph created in the applet?
How is your sketch similar to the graph created in the applet? Writing Equations  Using the data to the right, write an equation using the cosine function that represents your motion on the carousel for one complete revolution. Now try to rewrite the equation using the sine function. Check your work here. Graphing
 Graph each of the equations above which represents your motion on
the carousel during one complete revolution. Are the graphs the same?
If not, you should go back and rewrite your equations. Compare and Contrast  Now that you have completed the graph representing your motion on the carousel, how did you do? Compare your graph to THIS! 
Data About the Carousel The maximum height the horse reaches is 52 inches while the minimum height is 36 inches. It takes 14 seconds for the carousel to make one complete revolution. During this revolution the horse returns back to its maximum height four times. You should assume the horse starts and stops at its maximum height. 
Extended Carousel Investigations 
For further investigations, click here for a worksheet. Once you finish the worksheet, click here for solutions page. (Be patient, the solutions take a moment to load)

Take
a look here to see our work
in progress

July 25, 2002
Kara Fromke, Annette Hagelberg, Jami Stone, Ray Obst