Lab 2 Physics 190 Acceleration “g” Due to Gravity – Method 2 Introduction Tonight we will measure the acceleration due to gravity again. This time however, we will collect more data and the analysis will be different. We will first fit the data using a second order polynomial. Recall for a mass falling from rest, that 1 (1. 1) y ? a yt 2 2 Suppose a mass falls through n successively greater displacements, each time starting from rest. The displacements can be expressed a 2 y? ? y t? ; ?? ? 1 n ? . (1. 2) 2 Analyzing the Data Data for y? is not linear in time t?. We have two unique ways we can analyze the data.
The first is to simply plot the data with vertical displacement on the y-axis and time on the x-axis and perform a 2nd order polynomial curve fit. We can then extract acceleration from the coefficient of the 2nd order term. The second method involves transforming the nonlinear data into a linear form by means of the logarithm from which we can extract acceleration. We are going to use both methods because it demonstrates the power of mathematics as a data analysis tool. Fitting the Data to a 2nd Order Polynomial Free-fall data is shown in figure 1 and has the form y ? At2 ? Bt ? C (1. 3) Figure 1.
Free-fall plot (dots) and 2nd order fit (solid line). If we fit ideal free-fall data to equation (1. 3) we should find that B = 0, C = 0, and A = ay/2. If you look at the polynomial fit equation embedded in figure 1 you will see BWhitecotton Page 1 of 7 Lab 2 Physics 190 that B = -10-13, C = -10-14, and A = -4. 905. So the data is not perfect but essentially both B and C are zero while A = -4. 0905. If you compare the polynomial equation to our kinematic equation… y ? At 2 ? Bt ? C a y ? y t 2 ? vyit ? yi 2 …it becomes immediately evident that B corresponds to initial velocity, C the initial position, and A = ay/2.
If dropped from rest, initial velocity and position are zero. This all boils down to the fact that fitting a second order polynomial to free-fall data should provide the acceleration due to gravity directly. Simply plot displacement (yaxis) vs. time (x-axis) and use Excel, Vernier, calculator, or any tool that will perform a polynomial fit of order 2. Then ay = 2A which in the example above gives ay = 2(-4. 905) = -9. 81. Using the Logarithm to Linearize Data and Fit We begin with equation (1. 2), generalize and take absolute value ay m y? ? t? . 2 Vertical in figure Time Equation (1. 4) is plotted as data belowDisplacement vs2. 5 (1. 4) 20 |y(t)| (m) 15 10 5 0 0 0. 5 1 t (sec) 1. 5 2 2. 5 Figure 2. Absolute value of vertical displacement versus freefall time. Taking the log we obtain ? ay ? ?. log ? yn ? ? m log ? tn ? ? log ? ? 2 ? ? ? mXn Y n (1. 5) B Equation (1. 5) has the slope-intercept form of a line. Plotting the log of the data of figure 2, we obtain figure 3. The curve fits a straight line that has the form of Y = mX + B with m = 2. 0108 and B = 0. 6896. BWhitecotton Page 2 of 7 Lab 2 Physics 190 Linearized Data 1. 5 y = 2. 0108x + 0. 6896 R2 = 1 1 0. 5 Log( |y(t)| ) 0 -1. 2 -1 -0. 8 -0. 6 -0. 4 -0. 2 -0. 5 0 0. 2 0. 4 1 -1. 5 Log(t) Figure 2. Linearized data from figure 1 data above. Recalling that B = log(|ay|/2) = 0. 6896, we can solve for the acceleration ay. Inverting we get ay ? 100. 6896 2 ay ? 4. 893 . 2 a y ? 9. 787 Recall that our lab is at latitude ? = 32. 745°. Therefore the acceleration due to gravity in our lab should have magnitude g? ? 9. 795 . Computing experimental error we find ?a y ? g? g? ? ? 100% ? ?9. 787 ? 9. 795? ?100% ? ?0. 0863% . 9. 795 This is quite respectable but also uncharacteristically low for experiments in our lab. This experiment, if carefully done, can yield 1% error. BWhitecotton
Page 3 of 7 Lab 2 Procedure Physics 190 Set up the apparatus as we did last week. See figure 3 below for typical arrangement – this should look familiar. Spherical mass to= 0 s Digital Timer 0. 013s tf = t Figure 3. Setup for the free-fall experiment. You must complete 3 trials for each of 10 height settings. Use Table 1 to record data. Common Steps ? Set up the apparatus. ? ? Set the ball clamp to the first height y1 = 0. 53 m. ? Place the ball in the mount and measure the exact vertical displacement from the bottom of the ball to the compressed target mat. Please be sure to measure the displacement each time! Record the magnitude of y1 in Table 1 as your first of 3 trials. ? Make sure the timer is set in the correct mode and reset to zero. ? Release the ball and record the time of freefall in Table 1 as well. ? Repeat this procedure until columns |y? | and t? of Table 1 are complete. Polynomial Fit Steps ? Compute the means and record y? and t? of Table 1. ? ? Using your analysis tool of choice, plot y? vs. t? and label the axes appropriately. Fit a 2nd order polynomial to the mean data and instruct the tool to display the fit equation and the R2 value. You may need to omit a few of the lowest values if they are excessive outliers due to ? measurement uncertainty. This is legitimate when we understand equipment limitations. BWhitecotton Page 4 of 7 Lab 2 Physics 190 ? Compute ay from the 2nd order term: ay = _____________ m/s2. Show work here Log Method Steps ? Next, take log (use base 10) of y? and t? and complete the last two columns ? ? of table 1. Plot log( y? ) vs. log( t? ) and once again label the axes appropriately. Fit a 1st order polynomial (linear regression) to the data and instruct the tool to display the fit equation and the R2 value. You may need to omit a few of the lowest values if they are excessive outliers due to ? measurement uncertainty. This is legitimate when we understand equipment limitations. Obtain the y-intercept term B = log(ay/2). Compute ay from the y-intercept: ay = _____________ m/s2. ? ? Show work here Error Analysis Compute percent error for ay with respect to g? in the cases of the Polynomial Fit Method and the Logarithm Linearization Fit Method. Lastly compute the percent difference between the acceleration values determined from these methods. Questions 1. What are sources of error in this lab? 2. Why is it necessary to use the absolute value of the displacements when computing the log values? . Which of these methods gave the best results and why do you think that is? 4. What does the R2 value indicate when curve fitting to data? BWhitecotton Page 5 of 7 Lab 2 Formal Lab Report Physics 190 I want you to write a formal report on this lab. Follow the guidelines described in the formal report document available on my Cuyamaca homepage. Your focus should be on tabulation of data and the analysis (plotting of both raw and linearized data) including error analysis. Your final results should be emphasized and any error(s) discussed with thoughtful insight.
I want original work from each student with name and group name on the first page. Due ____________________ Logarithm Refresher Recall that the logarithm of an argument returns the exponent that operated on a base producing the argument. I know it sounds confusing. Let’s take a look. Suppose I had the number 1000. Well, 1000 is the same as 10 3. Here, 10 is the base and 3 is the exponent. If I operate on the value 1000 with the base-10 logarithm (denoted log10) like so, log10(1000), I obtain the result 3 which is the exponent that would operate on base-10 to produce 1000.
The operation can be expressed log10 ? 1000 ? ? log10 103 ? 3 ? ? There are many rules for using the logarithm. A few important ones for us are shown in the following examples… log ? k ? r ? ? log( k ) ? log(r ) ? d? log ? ? ? log(d ) ? log(b) . ?b? log c7 ? 7 log(c ) ? ? (See me or refer to the appendix in the back of the text if you need more help on logarithms) BWhitecotton Page 6 of 7 Lab 2 Table 1. Raw and processed data. Setup ??? : Positions 1: Set y ? 0. 53 m trial 1 trial 2 trial 3 mean 2: Set y ? 0. 66 m trial 1 trial 2 trial 3 mean 3: Set y ? 0. 9 m trial 1 trial 2 trial 3 mean 4: Set y ? 0. 92 m trial 1 trial 2 trial 3 mean 5: Set y ? 1. 05 m trial 1 trial 2 trial 3 mean 6: Set y ? 1. 18 m trial 1 trial 2 trial 3 mean 7: Set y ? 1. 31 m trial 1 trial 2 trial 3 mean 8: Set y ? 1. 44 m trial 1 trial 2 trial 3 mean 9: Set y ? 1. 57 m trial 1 trial 2 trial 3 mean 10: Set y ? 1. 70 m trial 1 trial 2 trial 3 mean Physics 190 Raw Data Polynomial Logarithm log( y? t? y? t? y? ) log( t? ) ? ? ? ? ? ? ? ? ? ? Use this table for data collection but make your own table in your report! BWhitecotton Page 7 of 7