Younghee Kwon PHYS 2125 Freely Falling Object Short Abstract: The purpose of this lab is to analyze one dimensional motion by looking at freely falling objects and determine the acceleration of gravity by timing the motion of freely falling objects. Two different methods involving picket fence and metal ball with different masses are used to measure the distance and falling time of the object. The results show constant and precise values for the gravity s acceleration due to minimized experimental errors. Based on the data collected from two experimental methods, theoretical values of acceleration can be determined by both calculation and distance vs. time graphs. Introduction: A freely falling object has motion in one dimension with an acceleration of gravity g = 9.8 m/s 2 in downward direction. Depending on the location and technical equipment, the value of the gravity s acceleration can be measured differently. The experiment involves two different methods. The first method uses the picket fence to pass through the photo gate which is already connected to a computer to measure the time accurately. The second method requires a person to drop two different sizes of balls at a certain height for several times. The time takes for each ball to reach the ground is measured by a stop watch controlled by one people. The data is then collected to calculate the acceleration of two different balls and the Excel is used to graph the relation between the distance and time for the balls to freely fall to the ground. In comparison of passing a picket fence through a photogate and measuring the time required to drop a ball, the computerized method is supposed to have more accurate measurement, thus yielding the value that is closest to the theoretical acceleration. However, our stop watch method appears to have closer value to the theoretical acceleration. The errors in measurement due to the reaction time seem to barely affect our experiment. The linearization equation can be found by making a linear, exponential, and graph of each data. The slope, m, and the intercept, b, is taken for the graph in which the R 2 value is closest to one. The slope and m are collected from Excel under the form of y = mx + b which is also known as a trend line; a line that best fits all the points. In order to find the experimental equation for an exponential graph, the natural log of the y intercept should be taken. The slope and intercept are then put into the y = be mx form. The experimental equation for a power graph can be found by taking the log of the y intercept and raising the x to the slope in y = bx m form. The result from the experimental equation represents the relation between the distance the ball travels and the time required to drop a ball.
Equations: Percent error = observed theoretical theoretical 100% The possible errors in measurements can be analyzed by comparing experimental and theoretical data. Mean (average value): x = N i=1 (x i ) N Average value is determined by the sum of total data divided by the actual number of experimental trials N. Standard deviation s (uncertainty): s 2 = N (x i x ) i=1 2 N 1 When a data is recorded by human being s observation, there is a potential possibility of several errors that might affect the result outcome. We can calculate this uncertainty by using sum x i, mean value x, and total number of trials minus 1. Standard error on the mean: SEOM= s When a person measures the time for the two metal balls with different masses to freely fall, she has to take reaction time into consideration as well. To get the reaction time for each distance and ball, one should take the average measured time for each distance and subtract from this the theoretical time one just calcualted. The reaction time for each person might be different, so it is important for one person to continuoulsy take the measurment throughout the trials. When two balls of different masses are freely fallng, the same value for acceleration should be acquired theoretically. A freelyfalling object has a motion in one dimension with an acceleration of gravity, g = 9.79 m/s 2. Since the mass doesn t affect the acceleration, the time taken for two balls with different masses to drop should be equal. N
Diagrams: Picket Fence 200 175 Photogate 150 125 100 75 (cm) Procedure: Picket fence experiment: Each group needs a picket fence, a photogate, a clamp, and a computer with LabPro Interface and Logger Pro. Set up the table clamp like described in the diagram and leave some spaces below for the picket fence for it to pass through. One person drop the picket fence through the photogate vertically as another person controls the start and stop timing process of the computer. Record the data into a given table. Stop watch experiment: Each group is assigned two metal balls with different masses, a stopwatch, and a height scale. One person will drop the ball at the indicated height for total of six trials, while the other person uses a stop watch to measure the time it takes for the ball to hit the floor. Record the date into a given table. Data: Table 1: Picket Fence Acceleration and Standard Deviation Trial 1 2 3 4 5 a (m/s 2 ) 9.889 9.887 9.695 9.871 9.825 Table 2: Data and Analysis for Mass 1 Height (cm) t 1 (s) t 2 (s) t 3 (s) t 4 (s) t 5 (s) t 6 (s) t ave_ball1 (s) 50 0.58 0.60 0.52 0.54 0.66 0.66 0.59 75 0.73 0.83 0.85 1.10 0.82 1.12 0.91 100 0.81 0.82 1.09 1.13 1.16 1.33 1.06 125 0.96 1.06 1.36 1.00 1.18 0.92 1.08
150 0.96 1.02 1.32 1.31 1.25 1.28 1.19 175 1.16 1.37 1.05 1.47 1.25 1.47 1.30 200 1.36 1.77 1.13 1.52 1.58 1.27 1.44 Table 3: Data for Mass 2 Height (cm) t 1 (s) t 2 (s) t 3 (s) t 4 (s) t 5 (s) t 6 (s) t ave_ball1 (s) 50 0.54 0.64 0.72 0.80 0.57 0.89 0.643 75 0.79 0.85 0.92 0.89 0.93 0.88 0.876 100 1.00 0.79 1.05 0.96 0.89 1.13 0.970 125 1.09 1.34 1.00 1.06 1.09 1.26 1.143 150 1.25 1.16 1.11 1.54 1.18 1.35 1.265 175 1.14 1.86 1.32 1.36 1.14 1.43 1.325 200 1.17 1.16 1.58 1.66 1.40 1.45 1.403 Table 4: Reaction Time Data 50 (cm) 75 (cm) 100 (cm) 125 (cm) 150 (cm) 175 (cm) 200 (cm) t ave_ball1 (s) 0.593 0.908 1.060 1.080 1.190 1.295 1.440 t ave_ball2 (s) 0.643 0.871 0.970 1.143 1.265 1.325 1.403 t theory (s) 0.320 0.391 0.452 0.505 0.554 0.598 0.639 t react_ball1 (s) 0.273 0.517 0.608 0.575 0.636 0.607 0.801 t react_ball2 (s) 0.323 0.485 0.518 0.638 0.711 0.727 0.764 t Ball 1 (s) 0.320 0.391 0.452 0.505 0.554 0.598 0.639 t ball 2 (s) 0.320 0.391 0.452 0.505 0.554 0.598 0.639 Calculations: g (average) = Standard deviation 9.889+ 9.887 + 9.695 + 9.871 + 9.825 5 = 9.833 (m/s 2 ) = (9.889 9.833)2 + (9.887 9.833) 2 + (9.695 9.833) 2 + (9.871 9.833) 2 + (9.825 9.833) 2 = 0.006651 (m/s 2 ) 5 1 SEOM = 0.006651 5 = 0.00297 (m/s 2 ) g ave ± g = 9.8334 ± 0.00297 (m/s 2 ) t aveball 1 = 0.58+0.60+0.52+0.54+0.66+0.66 6 = 0.59 (s)
t theory = 50 10 2 = 0.32 (s) 0.5 9.8 t reactball 1 = 0.59 0.32 = 0.27 (s) t Ball 1 = 0.59 (.273) = 0.32 (s) Percent error of stop watch experiment = 9.79 9.80 9.79 Percent error of picket fence experiment = 9.79 9.884 9.79 100% =.102% 100% = 1.01% Conclusion: The purpose of this lab was to understand one dimensional motion and the relationship between the distance and time of freely falling objects. Based on the results of the experiments and the graphs, the computerized equipment appeared to have slightly greater percent differences than the experiment that involved a human technique. (Q1,Q2) The graphs (distance versus time t Ball 1 (s) and t ball 2 (s)) for two balls with different masses yielded the exact same log and ln graphs. For the given theoretical equation, we expect the power graph to be a line since one of the constant acceleration formulas involves time squared which gives a graph that looks like a parabola. Since the usual log graphs also yields a shape of a parabola, our group thought the power graph (log graph) to be a line based on a given theoretical equation. Ideally, the power graph appeared to be a line as we expected since the R^2 value was exactly equal to 1. Y-intercept and the slope of the linearization equation can be easily found from the three graphs generated from the data tables. The slope and the y- intercept of the power graph were 2.0017 and 2.6904, respectively. The slope and the y- intercept of the exponential graph were 4.3359 and 2.5667, respectively. The experimental equation of the power relationship was found to be y=490.23x 2.0017. The experimental equation of this exponential relationship was found to be y=13.0227e 4.3359x. It seemed like there had been no discrepancies between theory and experiment when creating the graph since our R^2 value of power graph was exactly and precisely equaled to 1. (Q3) Our date gave approximate value of 9.79 for the g for both balls with different masses. This emphasized the fact that regardless of how big or heavy the ball is, the time of the two balls to freely fall should be the same. Through this little experiment, we could learn that the mass doesn t affect the acceleration of objects. However, in most practical situations, there is always a presence of air resistance. With air resistance, the ball with more mass would experience less effect of air resistance, thus fell a bit more rapidly than the one with the less mass. (Q4) In order to get the value of g from the ball experiment, we simply took the experimental equation of power graph (since it has the highest R^2 value) and set it equal to the theoretical acceleration formula given in the introduction,
Since the initial velocity of a ball is at 0m/s^2, we can eliminate the Vot. Before we set our power equation to the ½(at^2), we had to divide our power equation by hundred since it was originally calculated in centimeters, and we needed to convert to meter since the basic acceleration unit is in meters. Consider variable t^2 on the acceleration formula and x^2.0017 on the power equation same since they were very close numbers and thus canceled each other. At the end, we are only left with 2.9032 set equal to ½ a. solved for a, then we will get the value of g. The value of a and g should be the same since the only acceleration works against the freely falling objects is the acceleration due to gravity. The value of g from the picket fence can be easily calculated by simply calculating the g(average) from the data collected. It is odd that our stop watch experiment appeared to be more precise than the computer based experiment. Based on the percent difference, our stop watch experiment had significantly less percent difference than that of computer based experiment. (Q5) There are few possible sources of error that might have limited the precision in the result. For the ball experiment, slow and inconsistent reaction time, the ball not being dropped at the exact same height for every trial, and the ball not travelling a curved path instead of the straight downward direction might have been the possible sources of error. For the picket fence experiment, even though the method of picket fence experiment does not involve imprecise or slow reaction time due to the computerized techniques, there still might have been some errors. For example, the picket fence might not have dropped down straight when it went through the photogate.