Free Fall and Pendulum Experiment

Introduction:
The purpose of this lab was to learn the uncertainties in measurements, instrument uncertainty, random error, and total uncertainty. After learning how to do those, the best estimate value for the weights of the pennies over a period of 52 years was calculated.

In part A of the lab, a study is made between the amount of measurements made and the uncertainty of the value with the consideration of the mean shown in equation #1 below, standard deviation shown in equation #2, uncertainty due to fluctuation shown in equation #3, total uncertainty shown in equation #4, and the fractional uncertainty shown in equation #5 for each of the trials. All of the studies from part A were used to determine the weights of the pennies in part B.

x̄= (∑_(k=1)^N▒x_k )/N
Equation #1, Mean
σ= √((∑_(k=1)^(k=N)▒〖(x-x̄)〗^2 )/(N-1))
Equation #2, Standard Deviation
x_fl= σ/√N
Equation #3, Uncertainty due to fluctuation
∆x=√(〖x_ins〗^2+〖x_fl〗^2 )
Equation #4, Total Uncertainty
Fractional Uncertainty= ∆x/(x̄)
Equation #5, Fractional Uncertainty
The equations above were used to calculate the data in both Part A and B of the experiment.

Analysis:
Part A:
A workstation computer with the data acquisition (DAQ) card and a software that helped record the data from the experiment was used for all the parts of the lab. The software was used to input different measurements that were used to construct a histogram. The measurement that were inputted were also used by the software to construct a table that shows the trial number, the number of measurements, the mean, and the standard deviation. Then, equations 3,4, and 5 were used to find out the uncertainty due to fluctuation, total uncertainty, and the fractional uncertainty. Below are the calculations of trial 1 for the uncertainty due to fluctuations, total uncertainty, and the fractional uncertainty:

x_fl= 13.015/√3=7.5142
∆x=√(1^2+〖7.5142〗^2 )=7.58046

Fractional Uncertainty= 7.58046/100.16=0.07568

Trial No. of Measurements Mean Standard Deviation Uncertainty Due to Fluctuations Total Uncertainty Fractional Uncertainty
1 3 100.16 13.015 7.51421375 7.58046228 0.07568353
2 10 101.902 8.679 2.74454078 2.92104504 0.02866524
3 30 98.651 9.776 1.78484524 2.04589162 0.02073868
4 100 99.609 9.959 0.9959 1.4113174 0.01416857
5 300 100.347 9.613 0.55500681 1.14369251 0.01139738
6 1000 100.044 9.912 0.31344496 1.04797316 0.01047512
7 3000 99.792 9.921

0.18113185 1.01627198 0.0101839
8 10000 99.971 9.923 0.09923 1.00491124 0.01005203

Table 1: A2 Data
Figure 1: No. of Measurements vs Mean Value

The figure above shows the relationship between the amount of measurements made with the mean value. If more measurements are being made, then the mean value will be closer to the expected value of 100.
Error bars were added to the graph, as shown above. A horizontal line is added to each mean value on the scatter plot and the bigger the line the bigger the uncertainty is. A big difference is seen between the error bar for the first trial and the last. The error bars decrease in size, as more measurements are being made, which means the uncertainty due to fluctuation and total uncertainty decrease as more measurements are being made.

Part B:

Year Weight (g) No of Pennies Avg. Wt. (g) Std. Dev. Uncertainty due to fluctuations ∆wfl, Instrument Uncertainty Total Uncertainty Expected Wt. (g)
1959 3.091 1 3.091 0 0 0.001 0.001 3.1
1964 3.066 1 3.066 0 0 0.001 0.001 3.1
1966 3.094 1 3.094 0 0 0.001 0.001 3.1
1968 3.016 1 3.016 0 0 0.001 0.001 3.1
1969 3.109 1 3.109 0 0 0.001 0.001 3.1
1970 3.035 1 3.035 0 0 0.001 0.001 3.1
1972 3.084 2 3.0875 0.00495 0.0035 0.001 0.0045 3.1
1973 3.06 2 3.0705 0.01485 0.0105 0.001 0.0115 3.1
1974 3.087 4 3.07975 0.0221 0.01104819 0.001 0.0120482 3.1
1975 3.11 4 3.0895 0.02098 0.01049206 0.001 0.0114921 3.1
1976 3.079 1 3.079 0 0 0.001 0.001 3.1
1977 3.092 5 3.0966 0.02164 0.00967781 0.001 0.0106778 3.1
1978 3.052 4 3.07825 0.0221 0.01104819 0.001 0.0120482 3.1
1979 3.057 6 3.06983 0.0593 0.02420801 0.001 0.025208 3.1
1980 3.079 4 3.06775 0.06087 0.03043401 0.001 0.031434 3.1
1981 3.096 7 3.0989 0.0229 0.00865593 0.001 0.0096559 3.1
1982 3.096 4 3.09175 0.0135 0.00675 0.001 0.00775 3.1
1983 2.516 5 2.5134 0.01665 0.00744715 0.001 0.0084471 2.5
1984 2.488 6 2.5297 0.03039 0.01240609 0.001 0.0134061 2.5
1986 2.541 5 2.507 0.02887 0.01291124 0.001 0.0139112 2.5
1988 2.521 3 2.516 0.00723 0.00417665 0.001 0.0051767 2.5
1989 2.471 5 2.5066 0.02872 0.01284368 0.001 0.0138437 2.5
1990 2.489 7 2.4933 0.01339 0.00506018 0.001 0.0060602 2.5
1991 2.522 4 2.49675 0.02089 0.0104433 0.001 0.0114433 2.5
1992 2.505 4 2.48725 0.03385 0.01692323 0.001 0.0179232 2.5
1993 2.484 4 2.483 0.00852 0.00426224 0.001 0.0052622 2.5
1994 2.501 7 2.4917 0.01179 0.00445461 0.001 0.0054546 2.5
1995 2.492 4 2.493 0.0092 0.00460072 0.001 0.0056007 2.5
1996 2.481 2 2.472 0.01273 0.009 0.001 0.01 2.5
1997 2.497 9 2.493 0.01323 0.00441099 0.001 0.005411 2.5
1998 2.508 2 2.492 0.02263 0.016 0.001 0.017 2.5
1999 2.501 3 2.498 0.0052 0.003 0.001 0.004 2.5
2000 2.459 6 2.4865 0.01545 0.0063074 0.001 0.0073074 2.5
2001 2.501 5 2.495 0.01102 0.0049295 0.001 0.0059295 2.5
2002 2.49 2 2.4895 0.00071 0.0005 0.001 0.0015 2.5
2003 2.536 1 2.536 0 0 0.001 0.001 2.5
2005 2.491 3 2.4833 0.01686 0.00973539 0.001 0.0107354 2.5
2006 2.487 1 2.487 0 0 0.001 0.001 2.5
2007 2.497 2 2.499 0.00283 0.002 0.001 0.003 2.5
2008 2.502 1 2.502 0 0 0.001 0.001 2.5
2009 2.478 1 2.478 0 0 0.001 0.001 2.5
2010 2.496 5 2.4878 0.01809 0.0080895 0.001 0.0090895 2.5
2011 2.482 4 2.47525 0.01078 0.00539096 0.001 0.006391 2.5
Table 2: Data for penny weight from 1959 to 2011
In Part B of the lab, a digital scale with instrument uncertainty of 0.001 was used to weigh 150 pennies with production dates going from 1959 until 2011. One fact about pennies is that before 1983, pennies used to weigh 3.1 grams. In 1983 changes have been made and a penny weighs 2.5 grams since then. The mean weight of a penny in a given year was (Equation 1), the standard deviation (equation 2), the uncertainty due to fluctuations (equation 3), and the total uncertainty (equation 4) were all calculated for each year. Below is a sample calculation for year 1972:

x ̄= (3.084+3.091)/2=3.0875
σ= √((〖(3.084-3.0875)〗^2+〖(3.091-3.0875)〗^2)/(2-1))=0.00495
x_fl= 0.00495/√2= 0.0035
∆x= √(〖0.001〗^2+〖0.0035〗^2 )= 0.0045

Figure 2: Production year vs weight of coin
The scatter plot above shows the weights of coins produced in different years. The break or the gap in the graph is because there was a changed in the weight of a penny from year 1982 to year 1983, as weight changed from 3.1 grams to 2.5 grams.

The error bars on the graph should be representing the uncertainty for each year. There is a mistake in the graph above, as there should not be an error bar for years with only one penny trial, as the uncertainty would be 0.001, which should not be showing on the graph as it is too tiny.

Results:
Part A:
Part A also consisted of two quizzes. In part A1, the least count was 0.0010 and the instrument uncertainty was 0.001. In part A3, the best estimate was 1001, instrument uncertainty was 1.0000, uncertainty due to fluctuation was 0.8160, and the total uncertainty was 1.2910. Both of these quizzes had only one attempt and with the one attempt for each, two correct answers were achieved.
In part A2 as explained in the analysis section, uncertainty due to fluctuation and total uncertainty decrease with the increase in the amounts of measurements made. The more measurements, the closer the mean value gets to the expected value of 100.
Here are sample calculations of part B for year 1972:
x ̄= (3.084+3.091)/2=3.0875
σ= √((〖(3.084-3.0875)〗^2+〖(3.091-3.0875)〗^2)/(2-1))=0.00495
x_fl= 0.00495/√2= 0.0035
∆x= √(〖0.001〗^2+〖0.0035〗^2 )= 0.0045

Conclusion:
For part A, table 1 and figure 1 were constructed showing the relationship between the amount of measurements made and the closer the mean value gets to the expected value. The error bars in figure 1 show how uncertainty decreases, as more measurements are made, which is predictable. The decrease of uncertainty means the data collected is more accurate and precise with the increase in the amount of measurements made. With the 8th trial, 10,000 measurements were made, and the mean value was the closest of all to 100, which was 99.971.

For part B of the experiment, 150 pennies from 1959 to 2011 were weighed and based on weight standard deviation, uncertainty due to fluctuations, total uncertainty, and average weight were all calculated. Th expected weight from year 1959 to 1982 was 3.1 gram, while the expected weight of pennies from 1982 to 2011 was 2.5 grams. This was figured out after the experiment, as the weight of a penny has actually changed to save the government some money, as more zinc was used instead of copper, which weighs and costs less. The data was accurate, as the measurements made from 1959 to 1982 were close to the expected value of 3.1 grams, while the measurements from 1982 to 2011 were also close to the expected value of 2.5 grams. Therefore, this experiment was a success.