Phy31 Lab

testing ground 2 natural philosophy xcl Acceleration g bear onable to graveness method 2 Introduction this evening we resulting nib the acceleration receivable to gravity again. This clock time however, we will bundle more in socio-economic classation and the analysis will be different. We will first fusillade the info using a guerrilla swan polynomial. Recall for a draw falling from rest, that 1 (1. 1) y ? a yt 2 2 think a mass falls through n successively greater shifts, each(prenominal) time starting from rest. The displacements can be denotative a 2 y? ? y t? ? 1 n ? . (1. 2) 2 Analyzing the selective information Data for y? is not linear in time t?. We have two unique ways we can analyze the selective information.The first is to entirely dapple the data with vertical displacement on the y-axis and time on the x-axis and transact a second order polynomial wind gibe. We can then choice acceleration from the coefficient of the 2nd order term. Th e endorsementment method involves transforming the nonlinear data into a linear form by style of the lumberarithmarithm from which we can extract acceleration. We ar going to determination both methods beca usage it demonstrates the force play of mathematics as a data analysis pecker. conditionting 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 set(p) (solid line). If we fit ideal free-fall data to equality (1. 3) we should note that B = 0, C = 0, and A = ay/2. If you tactual sensation at the polynomial fit comparison embedded in figure 1 you will see BWhitecotton rascal 1 of 7 Lab 2 Physics one hundred ninety that B = -10-13, C = -10-14, and A = -4. 905. So the data is not perfect b arly essentially both B and C are zero objet dart 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 direct evident that B corresponds to initial amphetamine, C the initial position, and A = ay/2.If dropped from rest, initial velocity and position are zero. This all boils mound to the fact that fitting a second order polynomial to free-fall data should generate the acceleration repayable to gravity directly. scarcely plot displacement (yaxis) vs. time (x-axis) and use Excel, Vernier, calculator, or some(prenominal) tool that will perform a polynomial fit of order 2. Then ay = 2A which in the compositors case above gives ay = 2(-4. 905) = -9. 81. utilize the put downarithm to set Data and Fit We begin with equation (1. 2), generalize and take absolute jimmy ay m y? ? t? . 2 Vertical in figure measure Equation (1. 4) is plotted as data on a lower floorDisplacement 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 grow ? 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 bang-up line that has the form of Y = mX + B with m = 2. 0108 and B = 0. 6896. BWhitecotton pageboy 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 clear up for the acceleration ay. Inverting we get ay ? atomic number 6. 6896 2 ay ? 4. 893 . 2 a y ? 9. 787 Recall that our science laboratory is at latitude ? = 32. 745. T here(predicate)fore the acceleration due to gravity in our lab should have magnitude g? ? 9. 795 . cypher experimental phantasm we find ?a y ? g? g? ? ? 100% ? ?9. 787 ? 9. 795? ?100% ? ?0. 0863% . 9. 795 This is quite respec display board but also uncharacteristically low for experim ents in our lab. This experiment, if carefully done, can yield 1% error. BWhitecottonPage 3 of 7 Lab 2 Procedure Physics 190 batch up the apparatus as we did be week. See figure 3 below for typical arrangement this should look familiar. worldwide mass to= 0 s digital Timer 0. 013s tf = t Figure 3. setup for the free-fall experiment. You must complete 3 ladders for each of 10 height settings. Use get across 1 to magnetic disc data. Common steps ? Set up the apparatus. ? ? Set the swelling clamp to the first height y1 = 0. 53 m. ? Place the ball in the don and measure the exact vertical displacement from the bottom of the ball to the compressed orchestrate mat. Please be sure to measure the displacement each time destroy the magnitude of y1 in dishearten 1 as your first of 3 psychometric tests. ? addle sure the timer is set in the correct mode and reset to zero. ? consume the ball and record the time of freefall in Table 1 as well. ? extract this procedure until colu mns y? and t? of Table 1 are complete. Polynomial Fit Steps ? cast the hatchs and record y? and t? of Table 1. ? ? Using your analysis tool of choice, plot y? vs. t? and articulate the axes appropriately. Fit a 2nd order polynomial to the reckon data and instruct the tool to unwrap the fit equation and the R2 value. You whitethorn extremity to omit a a couple of(prenominal) of the worst set if they are luxuriant outliers due to ? measurement uncertainty. This is legitimate when we find out equipment limitations. BWhitecotton Page 4 of 7 Lab 2 Physics 190 ? Compute ay from the 2nd order term ay = _____________ m/s2. verbalise course here Log Method Steps ? Next, take log (use habitation 10) of y? and t? and complete the ending 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 a few(prenominal) 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 share error for ay with respect to g? in the cases of the Polynomial Fit Method and the Logarithm Linearization Fit Method. Lastly compute the part 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 orb Lab Report Physics 190 I want you to write a formal field of study on this lab. be the guidelines described in the formal repor t document available on my Cuyamaca homepage. Your centre 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 sort name on the first page. Due ____________________ Logarithm Refresher Recall that the log of an argument returns the exponent that operated on a petty(a) producing the argument. I know it sounds confusing. permits take a look. Suppose I had the number 1000. Well, 1000 is the same(p) 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) same so, log10(1000), I obtain the result 3 which is the exponent that would operate on base-10 to gain 1000.The operation can be verbalised log10 ? 1000 ? ? log10 103 ? 3 ? ? There are many rules for using the logarithm. A few important ones for us are s hown 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 textbook if you need more help on logarithms) BWhitecotton Page 6 of 7 Lab 2 Table 1. Raw and bear on data. Setup Positions 1 Set y ? 0. 53 m runnel 1 rivulet 2 ravel 3 dream up 2 Set y ? 0. 66 m foot race 1 trial 2 trial 3 incriminate 3 Set y ? 0. 9 m trial 1 trial 2 trial 3 correspond 4 Set y ? 0. 92 m trial 1 trial 2 trial 3 represent 5 Set y ? 1. 05 m trial 1 trial 2 trial 3 humble 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 BWhitecotto n Page 7 of 7