MATH13406 Applied Mathematics 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: =(19/sina) 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 m, as shown in the figure, the tension T is exerted in such a way that the system is in a counter clockwise impending motion state. T 14 MATH13406 Applied Mathematics Consider the mass m to be 4.0 Kg and the coefficient of friction to be 0.65. Write the equation connecting T and 6. Plot T 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 T in the rope? Take the natural logarithm from both sides of the equation. Explain that the equation you obtain is linear in In Tand, and therefore represents a line. What is the slope of the line? What is the y-intercept of the line? Plot In T versus on a Cartesian coordinate system. Show the slope and the y-intercept in the figure. Repeat items these for a general case, where the mass is m and the coefficient of friction is . You have been given a set of experimental values with this assignment. These values represent the tension T versus the angle of contact 6, 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 m 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 In T versus 0, rather than T versus 0, in order to use techniques for finding the line of best fit. From the data, create a table showing In T versus 8. 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 y-intercept of the line of best fit? What quantity does the y-intercept represent? Write down the numerical value of the mass m hanging from the rope, in appropriate units. Using the values of the coefficient of friction and the mass m, which you have already obtained, write down the equation for the tension 7 in terms of the angle of contact 0. 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 m, given a set of experimental data from an experimental set up used in this assignment. Comment on how this procedure can be modified to V-belt drives. This is the data for the tension in the string at different angles, for the same weight (m), based on the equation, T = me (radians) Tension (Newtons) 0 1961.33 0.25 2121.58 0.50 2294.93 0.75 2482.44 2685.27 1.5 3142.01