I need help with this lab. Please answer questions for pages 4-7.

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 = 9.80655 m/s2 are plotted below. Blue regions correspond to deviations of -5 x 10'4 m/szand the red regions correspond to deviations of 5 x 10'4 m 52. 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? Earth's Gravity Field Anomalies lmilligols] 60-40 30 -20 ~10 0 16 20 30 40 50 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 h in order to give a more accurate measure of the distance the ball will actually fall. Ignoring the height ofthe 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 ofthe 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 are = 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 v0 = 0 which leaves us with our theoretical model of freefall for this experiment: h ho = $th (1) We will be able to measure all of the values in the equation above except for 3. We will calculate g from the slope of a best fit line. If we plot t2 vs. h he 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 freebody 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 FM, 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: 1 FAir : EPACVZ '2) Just after the ball is released FM, 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 E45,, vector will grow large enough to match the Farm umber 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 ofair 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: 2 F : FGravity _ FAir : ma Carrying out the algebra we have the acceleration of the ball as: _mg FAir _ FAir a_g m m If FA\Data Table (small ball) h [711} MS] t2 [8] t3 [S] f [S] as [S] 1.9979 0.6599 0.6373 0.6388 1.7501 0.5944 0.6080 0.5989 1.5018 0.5620 0.5601 0.5565 1.2483 0.5062 0.5029 0.5246 1.0009 0.4475 0.4499 0.4513 0.7230 0.3894 0.3853 0.3918 0.4970 0.3155 0.3241 0.3283 Data Table (large ball) h [m] t1[3] t2 [S] 1'3 [S] f [8] Us [S] 1.9949 0.6256 0.6368 0.6285 1.7476 0.5959 0.5910 0.5840 1.4998 0.5489 0.5522 0.5526 1.2448 0.5021 0.5027 0.5015 0.9980 0.4500 0.4496 0.4430 0.7176 0.3811 0.3644 0.3699 0.4945 0.3176 0.3177 0.3002 Calculations (small ball) Calculations (large ball) h ho [m] f2 [s] 6'52 [5] h ho [m] f2 [s] 052 [s] i | | g small: % error = g large = % error = % difference = 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 1sigma error bars for f 2 and h ho and then we may use these as the bounds to find a maximum slope and a minimum slope for our data sets. We did just this in Lab 00. You will probably find, 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 gmin gs gmax J large ball [ gmin gr gmaxJ The brackets above represent a range of numerical values for the slope of our best fit lines. The central values g5 and g. represent the values of g 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 1sigma 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 ofthe two balls. (To the chosen standard at least...) We will assume that the standard deviation in the height measurement 0",, due to parallax is 2 mm or 0.002 m. This has not been measured or verified at all, but we will take this value as a reasonable but conservative estimate. In order to find the standard deviation in f 2 we need to use the propagation of error equation from Lab 00. Using equation (Al) on pg. 6 it can be shown that the standard deviation of f 2, based on our measurement of GE , is given by: 0'52 =250'g (3) 1. Calculate of 2. Calculate 052 according to equation (3) and let 03., = 0.002m 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 ofg are different? (in the 1sigma 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 t 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 actual/accepted value and xE is the experimental value, then lxx 'xsi Relative error = xx lxx 'Isi x 3. Percent error = x 100% This is used when comparing an experimental value to K 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 II = 3.16? Solution We found the error to be 0.02 in the previous example. The relative error is then E = 0.00637to a few decimal places. The percent error is then: 00063? x 100% = 0.64% Laboratory 00 Intro 4. Percent difference = %100%. When comparing two experimental values, the l 2 2 percent difference provides a measure of the precision of the experiment. Notice the denominator is the average of the values. 5. Personal errors are mistakes made by the experimenter when taking data or in calculating. 6. Systematic errors result from incorrectly calibrated equipment, poor laboratory habits, andfor incorrect zero point positioning. Repeating the measurement will not reduce the error. Systematic errors cannot be analyzed using statistics. 7. 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. 8. 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, 9.7 mfs2 or 9.5 mfsz? Solution 9.7 mfs2 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 significant gures (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 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 significant gures: a. The most signicant 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 significant 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 12004 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. 4Laboratory 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, 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 1 E ( x , - x )2 In-14 is a good measure of the precision of the measurements. 13. Standard error a = -" measures the precision of the mean. Vn 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 x ta 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: -_2.45m+2.47m+2.43m+2.51m+2.44m x = 2.46 m 5 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 :l: 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. Bumsgatinnnirm: 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 = kxaybz': the propagation will be found using the following equation: 2 (A1) R=a2"x +r;2 --* +c B) R = ax + by + 62 the propagation will be found using the following equation: (Bl) of? =a20': +2120": +c2r:rz2 mizL 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 nd 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 = m'4 (coefcient of the equation) x = (1 (rst measured value) b = 2 (exponent of the rst measured value) y = L (second measured value) c = l (exponent of the second measured value) 2 = 1 (no third measured value) (1 = 0 (no third measured value) Substituting the above into the matching error equation (A1), we nd: where the values of d and L would be the mean values. Solving for 0-,, yields the uncertainty of the volume. Smiths 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 ts, 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. Laboratory 00 Intro For questions (1) (8) use the following information. You have measured the length ofa table to be 205.0 cm, 205.8 cm, 205.4 cm, 204.6 cm, and 204.9 cm five independent times. You measured the width of the same table to be 60.1 cm, 60.4 cm, 60.2 cm, 60.0 cm, and 60.5 cm ve independent times. 1) Calculate the mean length L of the table. 2) Calculate the standard deviation of the mean length CL of the table. 3) Calculate the mean width W of the table. 4) Calculate the standard deviation of the mean width ow of the table. 5) Calculate the area A = L x W of the table. 6) Using the correct equation for propagation of error, calculate the uncertainty of the area (IA of the table. 7) Calculate the perimeter P = 2L + 2W of the table. 8) Using the correct equation for propagation of error, calculate the uncertainty of the perimeter Up of the table. 9) Report the mean length, mean width, area, and perimeter including their uncertainties in the correct format discussed above. Laboratory 00 Intro Gl'E' Suppose an experiment were conducted on the stretching of a spring as a function of the force applied to the spring yielding the data in the following table Spring extension sping extension standard deviation Force Force standard deviation (W) (cm) (N) (N) 0.00 0.06 0.0 0.5 1.25 0.07 1.0 0.5 2.08 1.04 2.0 0.5 3.87 0.32 3.0 0.5 4.31 0.40 4.0 0.5 5.62 0.79 5.0 0.5 6.42 0.25 6.0 0.5 6.60 0.79 ?.0 0.5 8.92 1.24 8.0 0.5 (Note: The values above are held to a few digits without consideration to the rules of reporting values discussed on pages 4 and 5. This is common for data and calculations tables to avoid roundoff errors in lture calculations. Following the rules for reporting values, the correct reporting of the value x in the third row of data above would be 2: = 2 :I: 1 cm and the correct reporting for row 4 above would be x = 3.9 :I: 0.3 cm, and so on.) First, you will need to decide where to draw the axes and what the scale should be. There are no absolute right or wrong choices, but some choices are better than others. Label the axes and mark convenient intervals Next, plot a graph of F vs. 3:. This means that F is on the vertical axis (ordinate) and x is on the horizontal axis (abscissa). Afx error bars to each point. Draw a straight line of length 20F, vertically and one of length 26",r horizontally centered on each point. Next, draw the theoretical curve. The theory states the relationship between F and x is a straight line through the origin. Draw the best straight line you can through the origin and the plotted points. Try to leave about as many points above and below the line. This is the line of best t. (Ask your instructor for the best fit ruler). Finally, draw the MAX and MIN lines. Draw two further lines through the origin with the largest and smallest slopes respectively that are reasonably near the plotted points. These lines should pass through the edges of some of the error bars. The MAX line will have a larger slope and the MIN line will have a smaller slope. 10 Laboratory 00 Intro Find the slopes of the three straight lines on your graph. You may use any two points on the line, but points well separated will provide better precision. To find the slope, nd the \"rise over the run\" , the change in vertical over the change in horizontal. Calculate the uncertainty of your line of best fit. Statistically, it is accepted to subtract the slope of the MAX line and the slope of the MIN line and divide by 2. This will give the uncertainty of the slope of the line of best t. A plot of the data with error bars is shown below. The slope of the best t is calculated to be 93 Nfrn. The slopes of the MAX line and MIN line, respectively, are 110 Nlrn and 82 N/'rn. The uncertainty of the line of best fit is then found by taking the difference of these two slopes and \"282 dividing by 2. This gives an uncertainty of = 14 me. Considering the rules of reporting significant digits, we write slope : 90 i 10 me . 11 Laboratory 00 Intro Force vs. Spring Extension F (N) x (cm) How do your hand-generated graph results compare? 12 \f9) The mean Length is 205.1 10.5 cm. The mean width is 60.21 0.2 cm. The area is 12 357.6 1 soc. The Perimeteris 530.8+1 (m