TL3. Graphical Analysis Prelab Checklist Intro Video(s) Prelab Quiz Introduction The purpose of an experiment is often to determine the value of a physical parameter. When determining this parameter, it is usually best to use data from several trials so that the value can be stated with confidence and the uncertainty is minimized. One way to do this is to use graphical analysis which also has the advantage of allowing us to visually assess our data. The first step for graphical analysis is to find a model (equation) relating the parameter of interest to directly measurable values. Then an experiment is designed such that the experimenter changes one of the measurable values (the independent variable), and the effects of this change on a second value (the dependent variable) are measured. By convention, data is usually graphed with the independent variable or some function of it on the horizontal axis and the dependent variable or some function of it on the vertical axis. To be able to determine the parameter of interest the equation relating the variables is linearized so that the resulting graph is a straight line with a slope (or sometimes intercept) that is related to the value of interest. A linearized equation is of the form y = mx+b. Note that the independent variable (the one you controlled) should be on the right side of the equation and the dependent variable should be the left side of the equation. Graphical analysis exercises: In this tutorial lab you will practice doing graphical analysis and linearizing equations in order to be able to determine values for particular parameters, - either from the slope or from the intercept of a resulting graph. The exercises you will be using for graphical analysis are: 1) Determining the density of glass marbles 2) Determining the spring constant (k) and Initial tension (IT) for a spiral spring: a) using Hookes Law b) using Simple Harmonic Motion (Note – this week you will be working on preparing the graphical analysis for the Spring Constant lab, and in week 4 you will be continuing to work on this system as we introduce the Experimental Design Sheet tool). You should begin working through the exercises now. 1. Density of Glass An experiment was designed to determine the density of the glass used to make marbles from a certain manufacturer by measuring the mass of marbles of various diameters. 𝑀 The equation that relates density (𝜌) to mass (M) is 𝜌 = 𝑉 where (V) is volume. In the case of a 4 𝑑 3 3 2 spherical marble 𝑉 = 𝜋 ( ) where d is the marble diameter. Combining these equations, we get 𝜌 = 4 𝑀 𝑑 3 𝜋( ) 3 2 . Simplifying this produces 𝜌 = 6𝑀 𝜋𝑑 3 . In the experiment, each marble diameter (d) is effectively ”changed” by the experimenter, and the mass (M) is measured. 1. What would the linearized equation be? Keep in mind that the x value on the right side should be a function of the independent variable (the one you change) and y value on the left side should be a function of the dependant variable (the one you measure). = y * m + x . b 1. What would you plot on each axis? (List both the symbol and name of the variable) Horizontal axis: Vertical axis: . . 2. Comparing the equation relating the variables to the equation for a straight-line y=mx+b write out an algebraic expression for the slope m of your graph m= . 3. Ρ = ___________ δρ = _________ 4. The mass and diameter data are given below. The diameter was provided by the manufacturer and is assumed to be exact. The mass has an error of 0.002kg. Diameter d [m] Mass M [kg] 0.017 0.007 0.022 0.015 0.025 0.019 0.035 0.054 0.042 0.102 0.050 0.169 This data table is also in the Excel spreadsheet provided. you need to plot to create your straight-line graph. Create a second table with the values Paste this table into the box provided below: Table 1: [insert appropriate caption here] Plot the data including error bars and the line of best fit and its equation. Insert the graph below. Figure 1a [insert appropriate caption here] Do a regression analysis to determine the error in the slope. Paste the residuals plot into the box below. m= ± . Figure 1b: [insert appropriate caption here] Qualitatively and quantitatively assess the graph and residual plot and comment in the box below. Does the line pass through about 70% of the error bars? Does the line pass through the origin within error – and do you expect it to? Based on the data, the line of best fit, and the residual plot – is the straight-line relationship a good model for the system? 5. Substitute in the numeric values into the equations in question 4 and solve for with error. = ± kg/m3 (number) 6. Look up the range of accepted values for the density of glass and the information below. Copy in your reference(s) = - kg/m3 (this value should be reported as a range) Reference website and date accessed: . Does your answer agree with the published values? Other comments on results? 2. Using Graphical Analysis to determine k for a spiral spring Next week you will be designing an experiment to determine the spring constant (k) of a spring as well as the Initial Tension of that spring – the tension that must be exceeded before the spring starts stretching. The experiment you design will involve using a graphical analysis to determine these values. Today you will be doing the preliminary analysis on this system in preparation for next week's experimental design. Hooke’s law and the equations governing simple harmonic motion (SHM) both provide models relating spring constants to the behaviour of springs. In what follows you will consider how an experiment could be done using each of these equations. At the end of today's lab you will choose which model you will use next week. A. The Hooke's Law Model Consider the schematic diagram in figure 1. When a load is applied to the free end of a spring suspended from a fixed support, the spring stretches a distance y until the tension in the spring, F, (also called the restoring force) just balances the load. Figure 1: Spring system free body diagram and coordinate system. Hooke's law states that F is proportional to the elongation y of the spring: The minus sign indicates that directions of F and y are opposite, i.e. the restoring force always points in a direction opposite to y. The quantity k is called the spring constant. The free body diagram for the system in Figure 1 shows that if the system is stationary then F = Mg. However, if the spring has a non-zero Initial Tension (IT) then there will be an offset in the force required to produce a given extension. For Mg > IT, This is the equation that you will be using for your graphical analysis. 1. In your experiment, what parameter would you vary? (Remember this is the independent variable – the one you change) 2. What would you measure? 3. Rearrange Equation 2 above to obtain a linearized equation for your experiment. Keep in mind that the x value on the right side should be a function of the independent variable (the one you change) and y value on the left side should be a function of the variable you measure. = y * m + x . b 4. What would you plot on each axis. Horizontal axis: Vertical axis: . . 5. Write out an algebraic expression for the slope m of your graph m= . 6. k= . 7. Write out an algebraic expression for the intercept b of your graph b= . 8. Rearrange this algebraic expression to solve for IT in terms of the intercept. IT = . B. The Simple Harmonic Motion Model Now you will develop a preliminary graphical analysis using a second model for a spring – Simple Harmonic Motion (SHM). The value of the spring constant is related to the period of the spring's oscillations during simple harmonic motion (SHM). A light uniform spiral spring with a mass M suspended at its free end will undergo SHM when set into vertical oscillations about a central equilibrium position (in other words: pull the mass down a bit and let go). The period T (time for one cycle) of the SHM is The Initial Tension will not affect the period but will limit the range of masses to those for which Mg> Initial Tension (in other words, you must chose masses that are heavy enough to stretch the spring). 1. In your experiment to determine k, what parameter would you vary? 2. What would you measure? 3. Rearrange equation (3) above to obtain a linearized equation for your experiment – so that "k" is contained in the slope m. Keep in mind that the x value on the right side should be a function of the independent variable (the one you change) and y value on the left side should be a function of the variable you measure. Write down your linearized equation below. = y * + m x . b 4. What would you plot on each axis. Horizontal axis: Vertical axis: . . 5. Comparing the equation relating the variables to the equation for a straight-line y=mx+b write out an algebraic expression for the slope m of your graph m= . 6. k= . C. Extension Question Discuss how the number of data points affects the reliability of your graphical analysis. Why is having at least six data points important, and why might ten data points be considered more respectable? How does increasing the number of data points help in reducing the impact of uncertainties and errors in the data? You will be designing a lab for next week using one of these methods. Decide with your lab group which of these methods you would like to use and indicate whether you will design your lab based on Hooke’s Law or Simple Harmonic Motion. Hooke's Law Simple Harmonic Motion NOW UPLOAD YOUR FILE TO THE ONQ DROPBOX FOR TUTORIAL LAB 3. IT MUST BE IN PDF FORMAT. TO DO THIS - BEFORE YOU LEAVE THE LAB GO TO “FILE” AND CLICK EXPORT. o CLICK “CREATE PDF” o SAVE IT (PUBLISH IT) TO YOUR TUTORIAL LAB 3 SUBFOLDER. THEN UPLOAD YOUR PDF FILE TO THE ONQ DROPBOX. DO NOT UPLOAD THE WORD FILE TO THE DROPBOX - MAKE SURE IT IS A PDF. WE WILL NOT MARK A WORD FILE.
