Phy31 Lab

Lab 2 physics one hundred ninety Acceleration g Due to Gravity Method 2 Introduction Tonight we impart evaluate the quickening out-of-pocket to gravity again. This clipping however, we will descend a line more information and the compend will be different. We will scratch stop the data using a second order polynomial. recollect for a mass falling from rest, that 1 (1. 1) y ? a yt 2 2 venture a mass falls through n successively greater displacements, separately quantify starting from rest. The displacements can be verbalised a 2 y? ? y t? ? 1 n ? . (1. 2) 2 Analyzing the Data Data for y? is not bilinear in time t?. We have two unique shipway we can analyze the data.The head start is to simply plot the data with unsloped displacement on the y-axis and time on the x-axis and per discrepancy a second order polynomial curve fit. We can then stir acceleration from the coefficient of the second order term. The second method involves transforming the nonlinear data into a linear form by take to bes of the lumberarithmarithm from which we can extract acceleration. We are going to use both methods because it demonstrates the power of mathematics as a data analysis apparatus. determineting the Data to a 2nd Order Polynomial Free-fall data is shown in mannikin 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 decide that B = 0, C = 0, and A = ay/2. If you look at the polynomial fit equation embed in figure 1 you will see BWhitecotton Page 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 but basically 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 dr opped from rest, initial velocity and position are zero. This all boils down to the occurrence that fitting a second order polynomial to free-fall data should pull up s seize ons the acceleration due to gravity directly. Simply plot displacement (yaxis) vs. time (x-axis) and use Excel, Vernier, calculator, or every tool that will perform a polynomial fit of order 2. Then ay = 2A which in the example to a higher place 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 care for ay m y? ? t? . 2 Vertical in figure Time Equation (1. 4) is plot 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. victorious the put down 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 fi gure 2, we obtain figure 3. The curve fits a dandy 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 science laboratory 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 kind of respectable but also uncharacteristically low for experiments in our lab. This experiment, if conservatively done, can yield 1% error. BWhitecottonPage 3 of 7 Lab 2 Procedure Physics 190 point up the apparatus as we did la st week. See figure 3 below for typical order this should look familiar. Spherical mass to= 0 s Digital timer 0. 013s tf = t Figure 3. launchup for the free-fall experiment. You must complete 3 streamlets for each of 10 height settings. Use Table 1 to record data. ballpark Steps ? Set up the apparatus. ? ? Set the ball clamp to the first off 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 propose mat. Please be sure to measure the displacement each time Record the magnitude of y1 in Table 1 as your first of 3 runnels. ? Make sure the timer is set in the temper 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 ? guess the mean(a)s and record y? and t? of Table 1. ? ? Using your analysis tool of choice, plot y? vs. t? and dog the axes appropri ately. Fit a 2nd order polynomial to the mean data and acquire the tool to display the fit equation and the R2 value. You may need to withdraw a few of the lowest values if they are excessive outliers due to ? criterion uncertainty. This is legitimate when we understand equipment limitations. BWhitecotton Page 4 of 7 Lab 2 Physics 190 ? account ay from the 2nd order term ay = _____________ m/s2. Show work here Log Method Steps ? Next, take log (use hind end 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). direct 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 disagreement 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 deliberation 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 breed on this lab. Follow the guidelines described in the formal report document functional 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 though tful insight.I want original work from each student with have and group name on the first page. Due ____________________ Logarithm refresher course Recall that the logarithm of an argument returns the world power that operated on a base producing the argument. I know it sounds confusing. Lets take a look. Suppose I had the number pace. 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 running play 1 streak 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