Cycloid Lab

A cycloid is the curve that is traced in space by a fixed point on the perimeter of a circle that is rolling along a flat surface. In this lab, we will investigate a mechanical property of cycloids. In particular, if a particle is moving in a constant force field, then the cycloid forms the path between two points that the particle will traverse in the shortest possible period of time. This often seems counterintuitive at first. Many people are inclined to believe that the straight line is the curve that will be traversed in the shortest period of time. We will not prove that the cycloid is the curve that can be traversed in the shortest possible time. This would require a background in variational calculus, which is beyond the scope of APPM 2350. However, we will show that it can be traversed faster than the line. In addition, we will also show that it can be traversed faster than a circular path. This will help support the notion that the cycloid is the curve that can be traversed in the shortest possible time.

I. Instructions

On your title page, clearly mark

• Names
• Student ID numbers
• Recitation number, TA name(s), and professor(s)
• If you do not include this information, as much as 10% will be deducted from your final score

For general directions for your report, please look at a sample report.

II. Goals

The goals with this lab are:

• Learn how to use computers in order to plot curves and solve equations numerically.
• Give practice in working with a project.
• Improve skills in report writing.

III. Mathematical Background

IIIa. Parametrization of a Cycloid

A common parameterization of the cycloid is:

Here, t is a dummy parameter. When t=0, we get x(0) and y(0) from the equations above. These are the coordinates of the initial position from where the cycloid will start being traced. As t increases, the cycloid is traced on the xy-plane, until t reaches its final value t_f. The value of a represents the radius of the imaginary circle that would be tracing the cycloid.

It would be a good idea for you to produce a few plots of the cycloid. This should give you a better geometric understanding of cycloids, which should in turn help you make more sense of the remainder of this lab. Any computer-based parametric plotter should produce a satisfactory picture. Try choosing several values for x(0), y(0), t_f, and a.

For more on cycloids, you can also read pages 738-741 in your Calculus book. If you still don't understand what a cycloid is, ask any TA or Lab Instructor.

IIIb. Parametrization of a Circle

The above parametrization traces a circle of radius r, centered at (c_x,c_y) on the xy-plane. For more on parametrization of circles, see page 735 of your Calculus book. 

In case you don't want to trace the full circle, but only part of it, you can do that by changing the initial and final limits of the parameter t. 

Hint: when solving this lab, you will need to use a quarter circle. Therefore, your parameter t won't start at 0 and won't end at 2pi.

IIIc. Parametrization of a Line

This was covered in Lecture, and can be found in section 10.5 of your Calculus book.

IIId. Length of a Curve

This was studied in Calculus 2. In case you need to refresh your memory, you may read section 5.5 and page 746 in your Calculus book.

For this lab, you will need to compute the length of parametrized curves. The formula needed to do so is:

IIIe. Traverse Time

The time a particle takes to traverse a given path under the force of gravity can be computed by the following integral:

Where g=9.81m/s² is the acceleration due to gravity.

IV. Problem Statement

The problem you are supposed to study and write a report with your results is the following:

Suppose you are a Civil Engineer that was hired by some internet book selling company to study 3 possible designs of a ramp that will connect the company's storage at the fifth floor of some warehouse to its loading dock on the ground. This ramp will be used to send books from the storage to delivery trucks. No energy will be required to move the books, since gravity will do all the work. But time is money, so the company would like to use the ramp design that takes the books from the storage to the loading dock in the least amount of time.

The storage room is located 13 meters above the ground, and the loading dock is positioned 13 meters to the right of the warehouse. Thus, you may consider the books to travel from point A=(0,13) to point B=(13,0) in the xy-plane. These are the points to be connected by the ramp.

The three possible shapes for the ramp are: 

• a straight line;
• a quarter-circle;
• a cycloidal arc.

With each curve connecting point A to point B, as can be seen in this sketch.

Given the above problem and the previous mathematical background, your task will be to plot all three shapes in the same coordinate axis, compute the length of each path, and compute the time required for the books to travel along each path, from point A to point B. Then, based on your results, you should state which of the studied paths is 'time wise' more efficient.

You should recall that the only forces acting on the books are gravity and the normal force from the path. You can neglect friction and air resistance. 

The following section will outline a few exercises, which should guide you solving this project.

The subsequent section will explain how you should write your report.

V. Lab Exercises 

1. Using the general parameterization of the cycloid that was stated at the beginning of this lab, find values for a and t_f  (recall that t₀ = 0) that correspond to the cycloid that connects A=(0,13) to B=(13,0). You will need to use a numerical rootfinder to do this. Numerical root finders need got initial guesses; as a general strategy, try plotting curves to understand the behavior and then propose reasonable geusses. In this case, (10,3) is a reasonable geuss for (a,t_f).
2. Find the coordinates (c_x,c_y) of the center of the circle and its radius r. Find the values for t₀ and t_f, which are the limits of your parameter t. These values should be chosen such that an exact quarter circle will join points A and B (this exercise does not require any computer calculations; you can do this by simply sketching the circle and understanding its parametrization equations).
3. Find a parameterization of the line segment between the points A and B.
4. Now that you have the values from exercises 1-3, you can use a parametric plotter to generate a picture of the cycloidal arc, the quarter circle and the straight line that connect points A and B. You should superimpose all paths in the same plot. Don't forget to label your axes and curves.
5. Calculate the lengths of the line segment, the quarter circle and cycloidal arc. You may be able to work some of these out by hand, but if you prefer or need, use some numerical integrator.
6. Calculate the time it takes for a book to traverse the line segment, the quarter circle and cycloidal arc. Once again, you may be able to work some of these out by hand, but if you prefer or need, use some numerical integrator.
7. Extra Credit: What are the final speeds of the books when they reach point B for each different path? Do they differ too much? In order to solve this question you may need to refer to your Physics book (hint: use conservation of energy. This is a very simple question). Make sure to state the equations you used, your results and comments.

The following links are to Mathematica notebooks applicable to this lab:

• Introduction to Mathematica
• Functions, Derivatives, and Plots
• Root Finding, Lists, and Parametric Plots

Mathematica has an informative introductory video and cyloid demonstration.

Organize the work you did in completing these exercises in a concise and understandable way. "What needs to be in your report?", you might ask. Read on.

VI. Lab Report

1. Your report should begin with an introduction. This should briefly describe what you plan to say in the body of your report. You should also provide a brief list of the mathematical concepts that you will use to make your arguments and perform your calculations.
2. The details of your work should be described in the body of your report. At the very least, you should discuss the following:
   • What is the 'real world' problem you are studying? What answers are you looking for?
   • What is the mathematical model of the 'real world' situation? Explain how the ramps are represented by the different parametrizations, what the starting and ending coordinates are. Tell which forces are acting on the books, and state your assumptions (frictionless ramp, no air resistance,...).
   • Provide a parameterization for (i) the straight line segment connecting the endpoints, (ii) the quarter circle connecting the endpoints and (iii) the cycloid connecting the endpoints. Be sure to explain how you arrived at these parametrizations. Don't forget to mention how you came up with and what were the initial guesses used to compute the parametrization of the cycloid. Include a plot showing all three paths superimposed connecting points A and B. This can be done in the body of your paper or may be referred to an appendix. Don't forget to label and refer to your plot.
   • State the lengths of the three curves. Show the integrals used in the computations, along with their results. Also, either show your integration work in the appendix or mention which software was used in computing the integrals numerically. Don't forget to use proper units in your results.
   • Also state the time it takes for a book to travel along these paths. Show the integrals used in the computations, along with their results. Also, either show your integration work in the appendix or mention which software was used in computing the integrals numerically. Don't forget to use proper units in your results.
   • Tabulate results when possible.
   • Compare the results obtained and state your opinion about which path is the most efficient. Can you come up with a physical reasoning that supports the results of your calculations?
   • Extra Credit: you don't have to do this part, but as a bonus, compute the final speed of a book at the end of each path. Compare them. State which equations from Physics you used and how Physics explains the speeds obtained. 
3. Finally, you should summarize what you have accomplished in a conclusion. No new information nor new results should appear in your conclusion. You should only review the highlights of what you wrote about in body. Briefly, what were you investigating? What answers did you get to? Are there any suggestions to better analyze/describe the same problem? 

The following link shows how your lab will be graded:gradesheet

Remember the Following:

• Always include units in your answers;
• Always label plots and refer to them in the text;
• The main body of your paper should NOT include lengthy calculations. These should be included in an appendix, and referred to in the main body.
• Labs must be typed. Including the equations in the main body (part of your learning experience is to learn how to use an equation editor). An exception can be made for lengthy calculations in the appendix, which can be hand written (as long as they are neat and clean), and minor labels on plots, arrows in the text and a few subscripts. 
• Your report doesn't have to be long. You need quality, not quantity of work. Of course you cannot omit any important piece of information, but you need not add any extras.
• DO NOT include print outs of computer software screens. This will be considered as garbage. You simply need to state which software you used in each step, and what it did for you.
• You must include any plot that supports your conclusions or gives you insight in your investigations.
• Write your report in an organized and logical fashion. Section headers such as Introduction, Background, Problem Statement, Calculations, Results, Conclusion, Appendix, etc... are not mandatory, but are highly recommended. They not only help you write your report, but help the reader navigate through your paper, besides giving it a cleaner look.    

During the weeks when the labs are out, there will be TAs present in ECCR 143 to help you with the labs. Please see lab hour schedule.