I need help with the highlighted questions. 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 = 9.80665 m/s2 are plotted below. Blue regions correspond to deviations of -5 x 10" m/szand 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 ofg 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? Eculh's Gravity Field Anomalies Imilligolsl 50-40 -30 -20 -1o 0 1E) 20 so 40 50 Materials: 0 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 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 1dimension to measure a value for g: 1 2 x=xo+ v0t+ 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 x0 = 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 v = 0 which leaves us with our theoretical model of freefall for this experiment: 11 ha = 29:2 (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 :2 vs. h h,J and calculate the slope of a linear fit to our data points then the slope of this line will be equal to $9. 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. FAI'T F Gravity The force due to air resistance is in the direction opposite to the velocity vector. It is also the case that PM, 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: Em = ipACvz (2) Just after the ball is released FAiT 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-T vector will grow large enough to match the Fmmy vector and the net force on the ball will become zero. With zero net force the ball will stop accelerating and the socalled 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: 2 F = Faravity _ FAir = ma Carrying out the algebra we have the acceleration of the ball as: "19 FAiT FAir a==g m m If FA\" were zero, the acceleration of the ball would be 9. 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 baiis 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 f 2. Square fto get f2 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 t line for each data set 5. Use slope = 3/2 to calculate an experimental value ofg for each ball 6. Calculate the percent error of g for each data set. Use the value of 3 out to four significant figures. Then g = 9.807 m/s2 (We have measured the time and height to 4 significant figu res) 7. Calculate the percent difference in the values ofg derived from the small steel ball and the larger and more massive steel ball ho = 0.0135 rn 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 mm Data Table {small ball) h [m] :1 [3] t2 [3] t3 [3] E [s] a; [5] 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 [5] t2 [3] t3 [5] E [s] a; [5] 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) 1111,, [m] f2 [s] 052 [s] h-h0|m| f2 |s| \"2 [5] 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 lsigma 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 gmin gs gmax l large ball l gmin gl gmax_ The brackets above represent a range of numerical values for the slope of our best fit lines. The central values g5 and gl 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 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 of the 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 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 (Al) on pg. 6 it can be shown that the standard deviation of f 2, based on our measurement of 0'5 , is given by: G'fz = Ztaf (3) 1. Calculate a"; 2. Calculate 052 according to equation (3) and let on = 0.002711 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 l-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