Project Number 1
Newton's Law of Cooling
The project consists of a word processed (typed) report submitted by the project report due date. Late submissions are penalized 10% of the mark per day, including Saturdays and Sundays.
The topic for the project is Newton's Law of Cooling, although you may choose a different title. The aim is to research how the temperature of a hot object is lost when it is kept in a cooler environment. You will need to use your mathematical skills, such as linear regression and knowledge of exponential functions. You will also be designing an experiment to find how water cools down from its boiling temperature. The project is also a good introduction on how to write a professional report which you will be faced with in your career. You should take care to ensure that all spelling and punctuation is correct and that the overall look of the report is polished and clean.
The report should consist of the following components:
1. A cover page. Include the Student Name, Course Code, Title and Date. Your cover page is the first thing the reader sees so it may be a good idea to add graphics to the page to draw attention to your report.
2. Introduction to the topic. Research the topic of Newton's Law of Cooling. Write the mathematical form of the law and explain its physical meaning. Try to justify the law using common sense and examples from all around you. Does Newton's Law of Cooling apply to convection, radiation, etc.? Briefly explain how, using Newton's Law of cooling, you may write the temperature of a cooling body as an exponential function of time. Is Newton's Law of Cooling still widely accepted? Are there any substitutes?
3. Description of the problem. Explain in detail how you would like to measure the rate of loss of temperature of water using an experimental set up. The experiment should be used to gather data for the temperature water versus time, as it cools down from its boiling point. Your descriptions and diagrams should be clear enough that someone new to the subject would be able to read your paper. One good measure of completeness is to ask yourself if you would be able to understand the equation by reading your description and studying your diagram. In other words your description should be a clear, self-contained exposition of the topic, much like one that you could find in the textbook on a given topic.
4. Description of the steps involved in solving the problem. Give a detailed description of the experiment and collecting the data. Mention that at least 15 data points should be collected on almost equal time intervals before the water reaches the environment temperature.
5. A detailed solution. Make a table of all of data you have gathered. Plot these data. What kind of function do you expect to see? Explain how using linear regression you can find the rate of loss of temperature for water. Apply this method, using Excel or other software, to your data and evaluate the numerical value of the rate of loss of temperature. Once done with that, write down the equation for the temperature of water versus time. Plot this equation and all of the data points on the same graph, and comment on the discrepancies you observe.
6. Conclusion and analysis of the results. Write a conclusion that highlights the major points of interest in your project and ties together some of the most interesting points you have covered.
Project Number 2
Heat loss in a Cylindrical Pipe
The project consists of a word processed (typed) report submitted by the project report due date. Late submissions are penalized 10% of the mark per day, including Saturdays and Sundays.
The topic for the project is Heat Loss in a Cylindrical Pipe, although you may choose a different title. The aim is to research a variety of pipes of different sizes made of different material, and a variety of insulators available in the market, and look at different combination which are economically feasible. The project is also a good introduction on how to write a professional report which you will be faced with in your career, you should take care to ensure that all spelling and punctuation is correct and that the overall look of the report is polished and clean.
The report should consist of the following components:
1. A cover page. Include the Student Name, Course Code, Title and Date. Your cover page is the fist thing the reader sees so it may be a good idea to add graphics to the page to draw attention to your report.
2. Introduction to the topic. Research the topic of heat loss in a cylindrical pipe. Explain which laws of physics are used to discuss heat loss in a pipe, and briefly explain how the famous equation for the loss of heat in a cylindrical pipe is derived. Explain the different variables and parameters involved, together with their appropriate units.
3. Description of the problem. Explain that your intention is to investigate a variety of combinations of pipe material, insulation material and size which you could potentially consider when deciding on pipeline system for a house. What other factors should be considered? What are the typical combinations that people use in Brampton? Is this different from what you find for other countries? If so, what are the reasons behind this difference? Your descriptions and diagrams should be clear enough that someone new to the subject would be able to read your paper and understand it. One good measure of completeness is to ask yourself if you would be able to understand the paper by reading your descriptions. In other words your description should be a clear, self-contained exposition of the topic, much like one that you could find in the textbook on a given topic.
4. Description of the steps involved in solving the problem. Outline in clear terms how you would find the necessary information, and how you would analyze, categorize and display the information you have gathered.
5. A detailed solution. List all the relevant types of pipes, insulators and sizes you have researched. Based on this information, calculate the heat loss per meter for each of the combinations you can figure out. Summarize the result in a graphical way, such as a table, such that they are easy to read and understand. Explain whether your findings for the best combination, which leads to the least loss of heat, matches what people use in your community. If not, elaborate on this. You may use a simple code to get the data and find all the heat losses for you.
6. Conclusion and analysis of the results. Write a conclusion that highlights the major points of interest in your project and ties together some of the most interesting points you have covered.
Project Number 3
Complex Variables and RLC Circuits
The project consists of a word processed (typed) report submitted by the project report due date. Late submissions are penalized 10% of the mark per day, including Saturdays and Sundays.
The topic for the project is Complex Variables and RLC Circuits, although you may choose a different title. The aim is to research how complex numbers facilitate some of the calculations in circuit theory. The analysis of several circuits can be summarized in a unified language, that is, the tool of complex variables. The purpose of the project is to shed some light on that. The project is also a good introduction on how to write a professional report which you will be faced with in your career, you should take care to ensure that all spelling and punctuation is correct and that the overall look of the report is polished and clean.
The report should consist of the following components:
1. A cover page. Include the Student Name, Course Code, Title and Date. Your cover page is the first thing the reader sees so it may be a good idea to add graphics to the page to draw attention to your report.
2. Introduction to the topic. Research the topic of RLC circuits with AC sources. Explain what is given in a general problem involving a circuit and what is asked for. Explain the terms which comprise an RLC circuit, with appropriate diagrams and details of the functions of each component, such as resistor, etc. Briefly outline the major historical figures involved in modeling the RLC circuits with AC currents, and explain ho and how started the use of complex variables in circuit analysis for the first time.
3. Description of the problem. State clearly that the current in an RLC circuit with an AC source with and without the use of complex variables. And that the purpose of your project is comparing the advantages and disadvantages of these approaches. Your descriptions and diagrams should be clear enough that someone new to the subject would be able to read your paper and understand the equation. One good measure of completeness is to ask yourself if you would be able to understand the equation by reading your description and studying your diagram. In other words your description should be a clear, self-contained exposition of the topic, much like one that you could find in the textbook on a given topic.
4. Description of the steps involved in solving the problem. Outline in clear terms the steps you would take relate the solution of the problem of finding currents in an RLC circuit with and without using complex variables.
5. A detailed solution. Using physical principles, explain how the current is obtained in an RLC circuit with an AC source. Explain how the phase difference between the current and voltage is obtained without the use of complex variables. Support your theoretical statements with an example. Explain how the same general problem can be solved using complex variables. Show that in both cases one obtains the same results.
6. Conclusion and analysis of the results. Write a conclusion that highlights the major points of interest in your project and ties together some of the most interesting points you have covered.
Project Number 4
Automotive Cruise Control System
The project consists of a word processed (typed) report submitted by the project report due date. Late submissions are penalized 10% of the mark per day, including Saturdays and Sundays.
The topic for the project is to design an Automotive Cruise Control System although you may choose a different title for your report. The aim is to research cruise control systems and the practical considerations involved in implementing a cruise control system in a vehicle. Essentially you will be designing a cruise control system of your own and facing some of the challenges that an automotive designer would encounter. The project is also a good introduction on how to write a professional report which you will be faced with in your career, you should take care to ensure that all spelling and punctuation is correct and that the overall look of the report is polished and clean.
The report should consist of the following components:
1. A cover page. Include the Student Name, Course Code, Title and Date. Your cover page is the first thing the reader sees so it may be a good idea to add graphics to the page to draw attention to your report.
2. Introduction to the topic. Research the topic of cruise control systems in a historical context and identify the complex mechanical and electrical hardware involved in implementing such a system. What are some of the difficulties that terrain, weather, safety and congested driving conditions introduce? What are some of the interesting modern developments in cruise control systems that contrast with comparatively basic old systems that may grab your reader's interest?
3. Description of the problem. Describe the cruise control system that you intend to implement along with the features you would like to include. What will set your system apart from other systems on the market and why will your system be preferred? Will your system be designed for high end vehicles or will it focus on economic applications?
4. Description of the steps involved in solving the problem. Identify the physical hardware and electronic or mechanical control devices that your system will include such as cameras and electronic throttle controls. What steps will need to be taken to take your proposition to production. What will be the most cost and time intensive components to design and manufacture, will any of the components be available as off-the-shelf products or will they need to be designed from the ground up?
5. A detailed solution. Create a detailed table of all the inputs and outputs your system will include along with the physical devices that will be required by each input and output, feel free to add other fields such as cost, supplier, and quantity. Make a complete and aesthetically appealing flowchart that details your systems operation, include graphics that will convince your boss, investor, or client that your system is the best choice. How will your system handle the logic involved in combining all of the inputs to produce a safe and predictable way of controlling the vehicle? Will you be able to make use of any mathematical functions or methods when you are processing the logic?
6. Conclusion and analysis of the results. Write a conclusion that highlights the major points of interest in your project and ties together some of the most interesting features you intend to include.
Project Number 5
Evaluating the reaction rate constant and initial concentration in first order reactions using experimental data
The purpose of this project is to show you the application of regression and exponential and logarithmic functions to a problem in your discipline. To be more precise, you will learn how to find physical quantities, such as reaction rate constants, from experimental data.
You may complete your project by going through the following steps. Introduction
Write the Introduction of your report by including discussion of the following items:
Describe what a first order chemical reaction is. Consider the reactants to be and .Indicate the relevant equations, including the equation showing the reaction and the equation describing the first order reaction.
Explain the following equation, including, above all, what represents:
Body
Write the Body of your report by including discussion of the following items:
Discuss an example of a first order reaction where the numerical values for and are given. Write an equation for as a function of time.
Plot versus on a Cartesian coordinate system. What happens to the concentration after a long time?
Take the natural logarithm from both sides of the equation. Explain that the equation you obtain is linear in and , and therefore represents a line.
What is the slope of this line? What is the -intercept of this line?
Plot versus on a Cartesian coordinate system. Show the slope and the -intercept in the figure.
Repeat these calculations for a general case, where the initial concentration is and the reaction rate constant is .
You have been given a set of experimental values with this assignment. These values represent the concentration of a reactant in a first order chemical reaction versus time, as measured in the lab.
The rest of the body of the project is about how to extract important information from these data, in particular, the initial concentration and the reaction rate constant . Here is the outline of how you can achieve this, and how you can present your work in a proper way.
Explain why you need to plot versus time, rather than versus time, in order to use techniques for finding the line of best fit.
From the data, create a table showing versus time. Plot these points on a diagram.
Using the method of least squares, find the equation of the line of best fit for these data points. Computing the correlation coefficient, explain how well the line of best fit represents the data. Plot the line of best fit on a coordinate system, as well as the data points.
Repeat previous steps using software to confirm your results.
What is the numerical value for the slope of the line of best fit? What quantity does the slope represent? Write down the value of the reaction rate constant with appropriate units.
What is the numerical value for the -intercept of the line of best fit? What quantity does the -intercept represent? Write down the numerical value of the initial concentration with appropriate units.
Using the values of the reaction rate constant and the initial concentration which you have already obtained,
write down the equation for the concentration at any time.
As some of the applications of this equation, show a couple examples how using this equation, you can find the concentration at a given time, and how you can find at what time the concentration is equal to a certain amount.
Conclusion
Write the conclusion to your report by including discussion of the following items:
State a general procedure how you would find the reaction rate constant and the initial concentration in a first order reaction, given a set of experimental data.
Project Number 6
Evaluating the time constant from experimental data in charging circuits
The purpose of this project is to show you the application of regression and exponential and logarithmic functions to a problem in your discipline. To be more precise, you will learn how to find physical quantities, such as time constant, from experimental data.
You may complete your project by going through the following steps. Introduction
Write the Introduction of your report by including discussion of the following items:
Describe capacitor charging in an RC circuit with DC current (charging circuit). Indicate the relevant equations, including the equation showing the current and voltage as functions of time. Explain the constants used in these equations.
Explain the following equation, including, above all, what and represent:
Body
Write the Body of your report by including a discussion of the following items:
Discuss an example where numerical values for and are given. Write an equation for as a function of time. Plot versus on a Cartesian coordinate system. What happens to the current after a long time?
Take the natural logarithm from both sides of the equation. Explain that the equation you obtain is linear in and , and therefore represents a line.
What is the slope of this line? What is the -intercept of this line?
Plot versus on a Cartesian coordinate system. Show the slope and the -intercept in the figure.
Repeat previous calculations for a general case, where the maximum current is and the time constant is .
You have been given a set of experimental values with this assignment. These values represent the current versus time in a charging circuit, as measured in the lab.
The rest of the body of the project is about how to extract important information from these data, in particular, the maximum current and the time constant . Here is the outline of how you can achieve this, and how you can present your work in a proper way.
Explain why you need to plot versus time, rather than versus time, in order to use techniques for finding the line of best fit.
From the data, create a table showing versus time. Plot these points on a diagram.
Using the method of least squares, find the equation of the line of best fit for these data points. Computing the correlation coefficient, state how well the line of best fit represents the data.
Plot the line of best fit on a coordinate system, as well as the data points. Repeat the previous steps using software, in order to confirm the results.
What is the numerical value for the slope of the line of best fit? What quantity does the slope represent? Write down the numerical value of the time constant in proper units.
What is the numerical value for the -intercept of the line of best fit? What quantity does the -intercept represent? Write down the numerical value of the maximum current in proper units.
Using the values of the time constant and the maximum current , which you have already obtained, write down the equation for the current in terms of time .
As some of the applications of this equation, show a couple examples how using this equation, you can find the current at a given time, and how you can find the time when the current equals a certain quanity.
Project Number 7
Evaluating the coefficient of friction in a flat belt-pulley system using experimental data
The purpose of this project is to show you the application of regression and exponential and logarithmic functions to a problem in your discipline. To be more precise, you will learn how to find physical quantities, such as the coefficient of friction, from experimental data.
You may complete your project by going through the following steps.
Introduction
Write the Introduction of your report by including discussion of the following items:
Describe V-belt drives and their applications. Discuss flat and V-shaped pulleys. Support your explanations with proper figures of the pulleys.
Using proper diagrams, and with a discussion of the variables used, discuss the following equation and its applications:
Explain the circumstance in which the above equation reduces to the following equation:
Body
Write the Body of your report by including discussion of the following items:
Consider the case of a flat pulley. Discuss an experiment in which for a certain angle and a certain mass , as shown in the figure, the tension is exerted in such a way that the system is in a counter clockwise impending motion state.
Consider the mass to be 4.0 Kg and the coefficient of friction to be 0.65.
Write the equation connecting and . Plot versus on a Cartesian coordinate system. If you keep on increasing the number of wraps of the rope around the pulley, what happens to the tension in the rope?
Take the natural logarithm from both sides of the equation. Explain that the equation you obtain is linear in and , and therefore represents a line.
What is the slope of the line? What is the -intercept of the line?
Plot versus on a Cartesian coordinate system. Show the slope and the -intercept in the figure. Repeat items these for a general case, where the mass is and the coefficient of friction is .
You have been given a set of experimental values with this assignment. These values represent the tension versus the angle of contact , as measured in the lab.
The rest of the body of the project is about how to extract important information from these data, in particular, the coefficient of friction and the mass hanging from the end of the rope. Here is the outline of how you can achieve this, and how you can present your work in a proper way.
Explain why you need to plot versus , rather than versus , in order to use techniques for finding the line of best fit.
From the data, create a table showing versus . Plot these points on a diagram.
Using the method of least squares, find the equation of the line of best fit for these data points. Computing the correlation coefficient, state how well the line of best fit represents the data.
Plot the line of best fit on a coordinate system, as well as the data points. Repeat the previous steps using software, in order to confirm the results.
What is the numerical value for the slope of the line of best fit? What quantity does the slope represent? Write down the value of the coefficient of friction.
What is the numerical value for the -intercept of the line of best fit? What quantity does the -intercept represent? Write down the numerical value of the mass hanging from the rope, in appropriate units.
Using the values of the coefficient of friction and the mass , which you have already obtained, write down the equation for the tension in terms of the angle of contact .
As some of the applications of this equation, show a couple examples how using this equation, you can find the tension for a given angle of contact, and how you can find at the angle of contact, given the force of tension.
Conclusion
Write the conclusion to your report by including a discussion of the following items:
Summarize the general procedure for finding the coefficient of friction and the mass , given a set of experimental data from an experimental set up used in this assignment.