According to a recent article in Popular Mechanics, the construction of roller coasters with loops is responsible for a resurgence of popularity for these amusements; loops are also implicated in several deaths. In this project, you will investigate some of the crucial aspects of roller coaster design.
One of the first companies to produce a coaster with a loop was Schwarzkopf GmbH of Germany. The so-called Schwarzkopf loop is described in the article as follows.
Sixty-three-year-old former cartwright Anton Schwarzkopf holds nearly 60 patents in the field of roller coaster design and manufacturing, and has produced more than 55 roller coaster systems around the world. His loop is the result of rigorous research and development conducted in the early '70s at his Bavarian test track. Other roller coaster makers had designed loops, but their ``geometry'' had imposed too many Gs on passengers, resulting in whiplash, broken collar bones, bruises , and other bodily strains. Schwarzkopf determined that a safe loop consisted of a spiral in which the radius of curvature decreased at a constant rate. Thus, most modern loops are tear-drop or oval-shaped, which means riders are subjected to slightly less than 6 Gs as the roller coaster's cars enter and leave the loop.
The key phrase in the passage above is ...radius of curvature decreased at a constant rate, at least as far as this project is concerned. In general terms, your first task is to find out what radius of curvature means and why it is important in coaster design. The second task is to investigate the Schwarzkopf loop. Can you come up with a model that retains the essential features, but is simple enough that you can solve it?
To do this project, you will have to read and understand the material in §14.4 and 14.5 in the text. This is where the unit tangent and normal vectors and of a curve are described, as well as the notion of a the curvature, of a curve.
In designing a roller coaster, the most important concepts are speed and force. You want the speed to be high, but not too high and the same goes for the forces on the passengers. According to the article quoted above, the total acceleration should not exceed 6 Gs. Obviously, we don't have a test track in Bavaria, so we will have to construct a simplified mathematical model. Here is the first set of assumptions.
Under these assumptions, Newton's second law and the decomposition
can be used to obtain the two scalar equations
Equation is a differential equation that will have to be solved for the speed function . Equation allows one to compute the normal acceleration, if you know the curvature and the speed . It wasn't made explicit in the equations, but , , , and all depend on time because the car is moving along the curve and , , and all change as the car moves.
Here are some specific things I want to see in your report, as part of the usual Introduction, Background, Procedure, and Conclusion structure. The next section of this handout provides hints and clues that should aid your investigations. You also might want to do some research in the library, including past MQPs.
Here are some suggested activities, designed to help you construct and analyze your own ``Schwarzkopf loop''.
where, hopefully, the function can be related to the curvature in some simple fashion. Try to generalize to a curve in three dimensions.
This is an open-ended project. That means, for one thing that there is no unique ``right'' answer. Here is what I think is a reasonable schedule for the completion of this project.