Please I need help with the questions for Calculations (Part ), Calculations (Part 2), the graph, and questions 1-2 on the last page. THE DATA TO DO THE CALCULATIONS ARE PROVIDED IN THE TABLES. THATS WHAT NEED TO BE SOLVED PLUS THE GRAPH. PLEASE SHOW WORK. PLEASE AND THANKS

Freefall Purpose: We will measure the value of 'little g', or the acceleration due to gravity on the surface of Earth. The particular surface location will be the LAVC Physics Laboratory space. The strength of the Earth's gravity varies by a small amount depending on location. The deviations from the average surface values of g = 9.80565 m/sZ are plotted below. Blue regions correspond to deviations of -5 x 10" m/s2 and the red regions correspond to deviations of 5 x 10\" m/sz. Once we have taken data and calculated a number, we will compare what we get to the known average surface value of g. We will also compare measurements of g using two different test masses with each other. Using the method developed in Lab 00 can we say whether our two measurements are different or not? Ecrlh's Gravity Field Anomalies |rnil|igolsl 60-40 -30 -2o -10 o 16 20 30 40 so Materials: 2-meter stick Photogate timer Free fall adapter for timer 2 steel balls of different mass and radius Electronic Balance Micrometer Spirit level Procedure: 1. Clamp your 2-meter stick to the laboratory bench. Check for vertical with the spirit level 2. Measure the height ha of the floor plate of the freefall adapter. This value will be subtracted from all of your meter stick height measurements It in order to give a more accurate measure of the distance the ball will actually fall. Ignoring the height of the plate would introduce a systematic error. 3. Choose 7 values for heights that range between 0.5m and 2m. Space them evenly. 4. Choose a height for the ball and set up the adapter and timer apparatus. Measure and record the height h from the underside of the ball to the floor 5. Release the ball. Record the timer value t that gives the time it took for the ball to fall from rest and hit the adapter plate on the ground 6. Repeat this measurement 2 more times for a total of 3 measurements for each height 7. Repeat these 3 measurements again using the other steel ball 8. Be sure to measure the height of the new ball to the floor again from the underside of the ball. The heights should be slightly different even though the ball holder in the freefall apparatus was not moved. The new ball will have a different diameter and will consequently settle into the freefall adapter differently. Theory: We will use the kinematic equation of motion in 1-dimension to measure a value for g: 1 2 x=xo+ vot+ Egt Note: The derivation of the above equation neglects the effects of air resistance. We are treating our steel ball as a point mass accelerating under the influence of a single, and constant, force. The gravitational force. We choose our coordinate system such that Jr = 0 and the actual distance travelled by the ball h ho is positive in the direction that the ball will fall. Since the ball is falling from rest, we also have that 120 = 0 which leaves us with our theoretical model of freefall for this experiment: h h. = $th (1) We will be able to measure all of the values in the equation above except for y. We will calculate g from the slope of a best t line. If we plot t2 vs. II h,J and calculate the slope of a linear fit to our data points then the slope of this line will be equal to g. This is because equation {1) is of the form y = mx where m is the slope of the line. Air resistance Below is the free-body diagram for a mass undergoing freefall. There is one force and the magnitude of the force remains constant throughout the fall. F Gravity If we consider the air resistance the steel ball will encounter, we have a free-body diagram with two forces acting on the ball. FAir F Gravity The force due to air resistance is in the direction opposite to the velocity vector. It is also the case that FAir is not constant. The magnitude of the air resistance is proportional to the velocity of the ball squared. A model for air resistance that is commonly used is below: Fm = lo/16v? (2) Just after the ball is released FAir is zero since the velocity v of the ball is zero. As the ball continues to fall the opposing force due to air resistance will increase as the ball speeds up. Eventually, if the ball falls for a long enough time, the FM, vector will grow large enough to match the Farm\"), vector and the net force on the ball will become zero. With zero net force the ball will stop accelerating and the so-called terminal velocity of the ball will be reached. In equation (2) the constant pis the density of the medium the object is falling through. In our case it is the density of air. The constant A refers to the cross-sectional area of our ball and C is a unitless constant whose value depends on the 3-dimensional shape of the object. This number quantifies how aerodynamic the object is. Less aerodynamic forms will have larger values of C. For a smooth sphere this value is around 0.42 Presumably, the effect of air resistance on our measurement of little g will be small since the velocities we reach will be small relative to the terminal velocity. The radius and hence the cross-sectional area A of each ball will be different. The mass of our experimental balls will also be different. Consider Newton's second law and our free-body diagram that includes air resistance. We get the following [F = FGravity - Fair = ma Carrying out the algebra we have the acceleration of the ball as: a = - mg - FAir = g - FAir m m If Fair were zero, the acceleration of the ball would be g. However, with a non-zero value of FAir , we see that a difference in the mass of the balls will matter for our two measurements of little g. In this lab exercise, we will not consider in any further detail the effect of the variables m, A, and v2 on the acceleration of the balls. We will only recognize that these variables do make some small difference. Our question will be - Can we measure a difference in the acceleration of the two balls given the precision of our experiment as determined by the sensitivity of our measuring devices and the procedure that we used? Calculations (Part 1): 1. Calculate t 2. Square t to get $2 3. Plot (h - ho) vs. 2 for each data set from each steel ball on the same graph 4. Determine the slope of the best fit line for each data set 5. Use slope = g/2 to calculate an experimental value of g for each ball 6. Calculate the percent error of g for each data set. Use the value of g out to four significant figures. Then g = 9.807 m/s2 (We have measured the time and height to 4 significant figures) 7. Calculate the percent difference in the values of g derived from the small steel ball and the larger and more massive steel ball ho = 0.0185 m Mass of small ball = 16.089 g ~ 16 g Diameter of small ball = 15.86 mm ~ 16 mm Mass of large ball = 27.753 g ~28 g Diameter of large ball = 20.03 mm ~ 20 mmData Table [small ball) h [m] r. [s] :2 [s1 :3 [s] f [s1 05 [51 1.9979 0.6599 0.6373 0.6388 I 1.7501 0.5944 0.6080 0.5989 I 1.5018 0.5620 0.5601 0.5565 I 1.2483 0.5062 0.5029 0.5246 I 1.0009 0.4475 0.4499 0.4513 I 0.7230 0.3894 0.3853 0.3918 I 0.4970 0.3155 0.3241 0.3283 Data Table [large ball) I h [m] :1 [5'] t: [s] t3 [5] f [5] 0f [5] 1.9949 0.6256 0.6368 0.6285 1.7476 0.5959 Calculations (small ball) Calculations (large ball) hha [m] P [s] hh [m] f? Is] Up [5] 3 small: % error = g large = % difference = % error = Calculations (Part 2): We have found the slopes of our best fit lines for both data sets. Now we will include the error in our measured data set to see if we can tell the difference between our measurements to the '1-sigma' level so to speak. That is, we will find the 1-sigma error bars for f 2 and h ha and then we may use these as the bounds to nd a maximum slope and a minimum slope for our data sets. We did just this in Lab 00. You will probably nd, if you took your data carefully, that this particular ball of larger mass and radius gave the larger value for little g. You can save yourself a little work by choosing which max/min slopes to calculate thoughtfully. The idea is to see whether the range of slopes for the balls, given the magnitudes of our error bars, will overlap. small ball 3min gs gmax l large ball l gmin gl gmaxi The brackets above represent a range of numerical values for the slope of our best fit lines. The central values gS and g represent the values ofg that you got from the results of Calculations (Part 1) for the small and large ball respectively. You can calculate the maximum and minimum slopes for each data set by either adding or subtracting the standard deviation of a measured value as described and practiced in Lab 00. If the range of slopes are overlapping, as diagrammed above (small ball gmax is greater than large ball gmin ), then we can say that our measurements are not different to the 1-sigma level. Our method and apparatus are not precise enough to distinguish a difference in the acceleration of the two balls. If we find no overlap, then we can say that we have measured a difference in the acceleration of the two balls. (To the chosen standard at least...) We will assume that the standard deviation in the height measurement on due to parallax is 2 mm or 0.002 m. This has not been measured or veried at all, but we will take this value as a reasonable but conservative estimate. In order to nd the standard deviation in f 2 we need to use the propagation of error equation from Lab 00. Using equation (A1) on pg. 6 it can be shown that the standard deviation of f 2, based on our measurement of of , is given by: afz = 21:65 (3) 1. Calculate a; 2. Calculate 052 according to equation (3} and let on = 0.002111 3. Find the maximum and/or minimum slopes you would need to check for overlap as indicated by your data and as described in Lab 00 4. Answer the Questions Questions: 1. Can you say whether or not your two measurements of g are different? {in the 1-sigma error sense..) 2. True freefall means that gravity is the only force acting on an object. Are we justified in neglecting the effects of air resistance for this measurement of g? *"'* Please attach two graphs to your report submission *** 1. A graph containing the best fit line of both the small and large ball data sets with fitting statistics displayed. 2. A graph showing maximum and/or minimum slopes for the small/large ball that tests for overlap. Fitting statistics should also be displayed on this graph. Laboratory 00 Intro Introduction to Measurements When making quantitative measurements that involve continuous variables, the level of uncertainty must be reported. Better instruments and laboratory procedures will yield results closer to the actual result. It is important to note that obtaining the exact answer is not as important as learning how to report a experimental value along with the level of uncertainty. In other words, you must be honest when reporting values. OBJECTIVES In this activity, you will interpret and analyze data. PART 1 RULES AND DEFINITIONS 1. Measurement error The difference between the experimental value of a quantity and the accepted value of the quantity. Error : Experimental value Accepted value Example 1 The accepted value of It to seven decimal places is 3.1415926. If a circumference experiment yields a value of It to be 3.16, then what is the error in the measurement? Solution Error = 3.16 3.14 = 0.02 2. Relative error The error of a quantity divided by the accepted value of the quantity. If xK is the actualiaccepted value and xE is the experimental value, then ilk3' Relative error = xx ixx _ xal xx 3. Percent error = x 100% This is used when comparing an experimental value to an accepted value. The percent error provides an idea of the accuracy of the measurement. Example 2 What is the percent error in the measurement of 7: = 3.16? Solution We found the error to be 0.02 in the previous example. The relative error is then 0.02 3.14 0.00637 X 100% = 0.64% = 0.00637to a few decimal places. The percent error is then: 4. Laboratory 00 Intro Ix:- xl|100%. When comparing two experimental values, the (x1 + x2) 2 percent difference provides a measure of the precision of the experiment. Notice the denominator is the average of the values. Percent difference = . Personal errors are mistakes made by the experimenter when taking data or in calculating. . Systematic errors result from incorrectly calibrated equipment, poor laboratory habits, and/or incorrect zero point positioning. Repeating the measurement will not reduce the error. Systematic errors cannot be analyzed using statistics. . Random errors that are produces by unpredictable and unknown variations. All personal and systematic errors can be eliminated, but some random errors will remain. Random errors can be analyzed using statistics. . Accuracy - The ability of a measurement to match the actual value of the quantity being measured or how close the measurement is to the true value. Example 3 If the actual value of gravity is accepted to be 9.8 mfsz, then which measured value is more accurate, 9.? mfs2 or 9.5 mfsz? Solution 9.7 er's2 is the correct answer since it is closer to the accepted value. 9. Precision The ability of a measurement to be consistently reproduced. The number of signicant figures (discussed below) in the reported value indicates the level of precision of the measuring instrument. Small random errors lead to higher precision. Example 4 Which group of measured values has a greater precision, (25 m, 26 m, and 24 m) or (22 m, 28 m, and 32 m)? Solution (25 m, 26 m, and 24 m) is a more precise grouping since the repeated measurement is closer in each case. Accuracy vs. Precision Consider the three images below. Ten shots are red at a target three separate times. Each shot is considered a single measurement. The goal is to hit the target's center. Laboratory 00 Intro Case 1 This data set is not precise (the repeatability of the measurements is low). None of the measurements are accurate, though the average of the data set may seem accurate (it may land near the center). Arbitrarily chosen measurements should have high percent difference. Without precision, the data set is not reliable. This is an example of using a tool beyond its limit or beyond the abilities of the user, such as ring too far from the target, or trying to measure the thickness of a mosquito's wing with a meter stick. Case 2 This data set is precise, but not accurate. The repeatability of the measurements is high (they are grouped closer together). The average of the data set is far from the center, though. Arbitrarily chosen data points will have low percent difference, but the average will have a high percent error. This is an example of a systematic error, such as incorrect sighting of the device, or not zeroing the tool properly. Case 3 i This data set is precise and accurate. The measurements are repeatable and the average is near the center. Arbitrarily chosen data points will have low percent difference, and the average will have a low percent error. 10. Signicant gures All the digits in a measurement that are certain plus one that is estimated. Rules for counting signicant gures: a. The most significant digit is the leftmost nonzero digit. In other words, zeros at the left are never significant. b. If there is no decimal point explicitly given, the rightmost nonzero digit is the least signicant digit. c. If a decimal point is explicitly given, the rightmost digit is the least significant Laboratory 00 Intro digit, regardless of whether it is zero or nonzero. d. The number of significant digits is found by counting the places from the most significant to the least significant digit. Example 9 - How many significant figures are in each value? Value Number of Significant Figures 232 3 23200 3 0.230 3 4.0012 5 2004 4 203.20 5 0.000030 2 Note that zeros can cause some confusion when counting significant figures. To clear this confusion, write potentially ambiguous values in scientific notation. Example 6 - How many significant figures does 8000 have? Solution - By the above method 8000 should have one significant figure. Example 7 - How can you report the value 8000 to have two significant figures? Solution - Rewrite 8000 as 8.0 x 10'. When measurements are added or subtracted, the answer can contain no more decimal places than the measurement with the left-most decimal place. When measurements are multiplied or divided, the answer can contain no more significant figures than the measurement with the fewest significant figures. Example 8 - 9.001 cm + 2.1 cm = 11.101 cm, but is reported as 1 1.1 cm, since 2.1 ends at the tenths place. Example 9 - 9.001 cm x 2.1 cm = 18.9021 cm2, but is reported as 19 cm2, since 2.1 only has two significant figures.Laboratory 00 Intro 1 1. Precision of the measuring tool - The smallest subdivision that can be read directly. If a single value is measured to be 25.0 cm with a tool of precision 1 mm = 0.1 cm, then the value should be reported as (25.0 + 0.1) cm. Reporting Values and Dealing With Random Errors 12. Mean and Standard Deviation - Random errors have an equal likelihood to be low or high compared to the true value. So, taking the mean x of many measurements X1, X2, ..., Xn is a natural way to reduce the effect of random errors. The mean is defined as x = -Ex n 1=1 and is the best value obtained from all the measurements. (Note: If several values are averaged, a general rule is to assign one more significant figure to the mean value.) Statistical analysis will show that the sample standard deviation ON1 = 1 [(x, - x) Vn-14 is a good measure of the precision of the measurements. 13. Standard error Vn On- measures the precision of the mean. 14. Reporting the uncertainty - The standard deviation (or standard error if many measurements are made) will substitute as the uncertainty for the mean of many measurements. It is necessary to report it correctly to the reader. Use the following format xto, , or xta It is important to note that it is necessary to keep no more than one significant figure in the standard deviation and the standard error. (Some texts will say that the standard deviation and standard error should be no more than two significant figures.) Be sure to keep the same decimal place in the mean as in the standard deviation and standard error, even if this means rounding the mean to a lower decimal place (you can remove certainty to ensure the decimal places match). Never add digits to the mean in order to match the decimal place of the standard deviation and standard error (you cannot add certainty). If the standard deviation or standard error is too small, then use the precision of the measuring tool. Example 10- Given the following measurements find the mean and standard deviation and report it in the correct format. 2.45 m, 2.47 m, 2.43 m, 2.51 m, 2.44 m. 5Laboratory 00 Intro 1 1. Precision of the measuring tool - The smallest subdivision that can be read directly. If a single value is measured to be 25.0 cm with a tool of precision 1 mm = 0.1 cm, then the value should be reported as (25.0 + 0.1) cm. Reporting Values and Dealing With Random Errors 12. Mean and Standard Deviation - Random errors have an equal likelihood to be low or high compared to the true value. So, taking the mean x of many measurements X1, X2, ..., Xn is a natural way to reduce the effect of random errors. The mean is defined as x = -Ex n 1=1 and is the best value obtained from all the measurements. (Note: If several values are averaged, a general rule is to assign one more significant figure to the mean value.) Statistical analysis will show that the sample standard deviation ON1 = 1 [(x, - x) Vn-14 is a good measure of the precision of the measurements. 13. Standard error Vn On- measures the precision of the mean. 14. Reporting the uncertainty - The standard deviation (or standard error if many measurements are made) will substitute as the uncertainty for the mean of many measurements. It is necessary to report it correctly to the reader. Use the following format xto, , or xta It is important to note that it is necessary to keep no more than one significant figure in the standard deviation and the standard error. (Some texts will say that the standard deviation and standard error should be no more than two significant figures.) Be sure to keep the same decimal place in the mean as in the standard deviation and standard error, even if this means rounding the mean to a lower decimal place (you can remove certainty to ensure the decimal places match). Never add digits to the mean in order to match the decimal place of the standard deviation and standard error (you cannot add certainty). If the standard deviation or standard error is too small, then use the precision of the measuring tool. Example 10- Given the following measurements find the mean and standard deviation and report it in the correct format. 2.45 m, 2.47 m, 2.43 m, 2.51 m, 2.44 m. 5Laboratory 00 Intro Solution The mean is: 2.45 m+2.47 m+2.43 m+2.51m+2.44m x: 5 =2_46m The standard deviation works out to be 0.0316 m. The correct form for reporting is: 2.46 :I: 0.03 m since the uncertainty is rounded to one significant digit (0.03) and the mean is rounded to match the decimal place. In this case, 2.46 ends at the hundredths place, which matches 0.03. If the uncertainty was calculated to be 0.3, for instance, then the correct reporting would be: 2.5 :I: 0.3 m. Sometimes the standard deviation will be calculated to be too small and will seem to be zero. In this case, we must use the precision of the measuring tool and the measurer's technique to estimate the uncertainty. In other words, the uncertainty would be the smallest value that the measurer can read directly. anagannnirm: It is not entirely trivial how to include the uncertainties in calculations involving more than one quantity with an uncertainty. It is important to use the method of propagation of error. There are two forms of equations that will be discussed here. A) R = iq\" ybzc the propagation will be found using the following equation: 2 2 2 2 0' 0' 0'. 0' (A1) =a2 \"b? -;+c2 ; x y z B) R : ax + by + (:2 the propagation will be found using the following equation: (B1) 0'; =azo': +1172cr:+czcrz2 ndZL Example 11 The equation for volume of a cylinder is V = , where d = diameter and L = length are the only two measured values. Since the V = volume is to be calculated from measured values, to find the uncertainty of the volume, you must propagate the error. Laboratory 00 Intro Solution The volume equation matches the form of equation A, since the measured values are being multiplied. The rst step is to put the volume equation in the form of equation A: R = V (calculated value) a = 1:14 (coefcient of the equation) x = d (rst measured value) b = 2 (exponent of the rst measured value) y = L (second measured value) c = 1 (exponent of the second measured value) 2 = 1 (no third measured value) d = 0 (no third measured value) Substituting the above into the matching error equation (A1), we nd: 2 2 2 0', 0' 0' 1 =22d+12 V2 d2 L2 where the values of d and L would be the mean values. Solving for 0-,, yields the uncertainty of the volume. (instills Several experiments will require you to construct a graph or a curve. Unless otherwise specied, these are to be done by hand into your notebook. The following items should be considered: 1. The Axes The horizontal axis is known as the axis of the abscissa, and the vertical axis is known as the axis of the ordinate. In most cases, the instructions should illustrate which quantities are to be plotted horizontally and which are to be plotted vertically. Generally, the independent variable, typically horizontal, is taken as the abscissa and the dependent variable, typically vertical, is taken as the ordinate. Often you are asked to plot the ordinate versus the abscissa. For example, if you are asked to plot F vs. 3:, then you construct a graph with F on the vertical axis and x on the horizontal axis. Always label the axes with the variables and their units. 2. The Table A table of quantities to be plotted should be made for convenience in plotting and to aid in selecting a scale. The units of each quantity should be identied at the top of the column. 3. The Scale The scale of the graph is the number of units that correspond to one space or block on the graph paper. The scale should be chosen for both ordinate and abscissa that the curve, when drawn, will extend over most of the paper. Remember that the larger the space in which the data fits, the more precisely the points can be plotted. At the same time, a convenient scale should be chosen which is not awkward. Consult your table to determine a suitable range. Before deciding on a scale, try it out to see if points can be plotted easily. In almost all cases, values should increase from left to right and from below to upward. Indicate the scale plainly by numbering the divisions. You do not have to number every division. Laboratory 00 Intro . The Plotted Points The plotted points should be small but identiable. If several curves are to be drawn on the same set of axes, use different identication around the points of each curve, circles for the first, triangles for the second, squares for the third, etc... This should be done before attempting to draw the curve itself. . The Curve After the points have been plotted, a curve corresponding to the theoretical expectations should be drawn. If, for example, a straight line is expected, it should be drawn in such a way that about one half of the points miss the line on the same side. This is the line of best fit. A linear least squared fit of the data can be performed. Use a ruler or straight edge to draw a straight line. _ The Title Write a title above the graph and caption below it if' necessary