The goal of thisexperiment was to find the errors associated with uncertainties duringmeasurement of the given objects. During this experiment,the lengths and masses of objects were measured using the basic equipment. Based on the equipment used, theprecision of the measurement changes. However, the measurements cannot be exactno matter how precise the equipment can be. In the first investigation, wemeasured mass, diameter and length of 4 metal cylinders in order to calculatetheir density and volume using the equations: (1.1) (1.
2) The unit for mass is in (g), the densityin (g/cm^3), length and diameter in (cm), and volume in (cm^3). After the density was calculated using the equations (1.1)and (1.2) the results were compared to the data we retrieved from the plot ofmass vs volume in order to check for accuracy. In the second investigation, a Geigercounter was used to measure the number of background radioactive counts presentin the lab. This was performed in order to practice error calculations.
Thedata was collected, analyzed and random errors were calculated for eachmeasurement and calculation. Investigation 1 The setup for this investigation consists of a digital scaleand graduated cylinder which was used torecord and compare the mass of 4 metal cylinders. The length and diameter ofeach cylinder were measured with a ruler.
All this equipment was placed on the working desk in order to perform eachstep. Each cylinder was numbered from 1-4. They were placed onthe digital scale respectively after it was tared and the mass was recorded intable 1.1. Error in mass ?m(g) was calculated on the second step of theprocedure.
The error estimate was ½ of the smallest increment of the digitalscale, which was ½ of 0.1g equal to 0.05g. the error in mass ?m(g) was thencalculated which was done by taking the ratio of ?m(g) and mass ?m/m. The third step consisted of measuring the diameter (D)and length (L) of each cylinder and determined the errors for both thesemeasurements. Again, the error estimate was ½ of the smallest increment on theruler.
In our case, both the absoluteerror in length (?L) and absolute error in diameter (?D) were ½ of 0.01cm equalto 0.005cm. After the errors were calculated the relative error of legth (?L/L)and relative error of diameter (?D/D) were calculated for each of the cylindersand they were inputted in Table 1.1 in the appropriate rows.
We proceeded incalculating the volume (V) using the measurements of the diameter and lengthand inputting them in the equation V=(?D^2L)/4. We then proceeded in measuring the error of each volume that wasmeasured using the formula: Afterwards, we used a graduated cylinder to record the volume of thelargest cylinder (#4) so we could compare it to the volume we calculated withthe formula in order to determine which volume was more precise and had lesserror.Next, the density of each of the 4 cylinders was calculated as well asthe relative error. In order to calculate the density, the mass and the volumecalculated in the previous steps were inputtedin the equation (1.1). Average density ( ) was calculated usingthe equation: and the relative error of the average density ( ) was calculated using the formula: Next, we proceeded tobuild a graph with mass(g) in the y-axis and volume (cm^3) in the x-axis inorder to achieve a better value for the average density and its relative error.The values were plotted as shown in Figure 1.2.
The slope of the graph was usedto calculate again the average density of the cylinders and the relative errorsince it is said to be a more precise way and used for comparison. Cylinder #1 #2 #3 #4 m(g) 14 22.1 39.5 73.7 ?m(g) 0.05 0.05 0.05 0.
05 ?m/m 0.0036 0.0023 0.0012 0.
0007 L (cm) 5.1 3.5 3.5 6.8 ?L (cm) 0.0005 0.
0005 0.0005 0.0005 ?L/L 9.80392E-05 0.000142857 0.000142857 7.
35294E-05 D (cm) 0.6 0.9 1.
2 1.2 ?D (cm) 0.0005 0.0005 0.
0005 0.0005 ?D/D 0.000833333 0.000555556 0.000416667 0.
000416667 V (cm^3) 1.441991028 2.226603793 3.958406744 7.
690618816 ?V (cm^3) 2.91445E-07 4.4355E-07 6.8289E-07 9.51856E-07 ?V/V 2.02113E-07 1.99205E-07 1.
72516E-07 1.23768E-07 ?(g/cm^3) 9.708798271 9.925429961 9.978762305 9.583104008 ?? (g/cm^3) 0.034951674 0.022828489 0.
011974515 0.006708173 ??/? 0.0036 0.0023 0.0012 0.
0007 Average Density 9.799024 (g/cm^3) Error of average ? 0.010986 Density from graph 9.52 (g/cm^3) Error of slope from graph 0.
01 Table 1.1- Measurementand calculations for massg, diametercm, lengthcm, volumecm^3, densityg/cm^3and their relative errors for Investigation 1 In order for the densityand volume to be calculated the mass(m), length (L) and diameter (D) of 4 metalcylinders was measured. Each of these measurements were used to plug inequations (1.1) and (1.2) and from these calculations the values of the densityand volume derived and were inputted in the table above. Figure1.2-Plot of mass vs Volume.
The error ofslope is equal to 0.01.The density and error calculated with IPL from thegraph values was 9.52 0.01g/cm^3.The graph portrayed in Figure 1.
2 was used to calculate a more precise errorand density from the slope. The values that were retrieved from the graph wereused for comparison in order to see how accurate our calculations using thevalues from the table were. The error bars are not portrayed in the graphbecause the error was very small it was 0.01. In comparison to the data thatwas achieved from the table the slope of the graph (density) the values werenot equal to the range of the random errors. This could be due to the roundingup our calculations and slightly wrong mass(m) measurements.Investigation2To set up thisinvestigation the GMC-200 box was connected to the power cable. One cable wasinserted in the Data input and then it was connected to the computer.
Thecounter was turned on by pressing the red button on the left side of thedevice. To begin with, the GQ GMCounterPRO software was started in order to collect the data that was needed. Then weproceeded to set up the software by going to Setting in the top menu, pressedthe Geiger counter option and GMC-060/080/100/200 with Audio Cable was selected.Before we started to collect the appropriate data, we had to go to File andselect Restart Counter.
This step made sure all previous data was erased. Nextthe program was set to automatically record the number of counts for 1 hour. Thefirst 20 entries were inputted in table 1.2.
To calculate random error of thesedata a histogram was put together. The number of bars needed for the histogramwas 10 and the bin size calculated was approximately 2. Trial Counts/60s Bins 18 12 12 19 15 14 1 16 16 16 16 18 13 17 20 10 19 22 15 19 24 6 20 26 7 20 28 11 20 30 17 20 2 21 3 22 4 22 14 22 8 23 5 24 9 28 12 30 20 30 Table1.2-Measurements of counts/60s forInvestigation 2.
(Data sorted by time)The data from the table was used to calculate theaverage value of background counts per minute (n). This was done by taking theaverage of all the values in the counts/60s column. The n value was equal to 20.8 counts/60s.
This value was comparedwith the value of another group and both values were similar but not equal. Figure1.3– Number of background counts per minute.The Histogram was used to estimate the error of theaverage count. This was calculated by theformula: Win this equation stands for the width that exists between the blocks that are ½of the height of the tallest block in the histogram and in our case ½ of thetallest block was the block with frequency 3. The error for the average countwas equal to 3.39 counts/60s.Conclusion Experiment1 consisted of two investigations in which random errors of measurements werecalculated.
We used different methods to compare the accuracy of each one ofthem in finding the value of density or volume. We also saw how everymeasurement is not very precise and we always have to take into account manyfactors. Ininvestigation 1 the mass, diameter and length were measured and their randomerrors. From these measurements the density and volume were calculated. Firstly,we calculated the value of the volume for each cylinder. The volume of thelargest cylinder (#4) was calculated via the equation but also with a graduatedcylinder. The volumes were slightly different. The volume we recorded from ourcalculations was 7.
69 1.23E-07 cm^3 and the volume from the graduated cylinder was 9.0 0.05 cm^3. This changecould be due to miscalculations and wrong measurements. Next, the density ofthe 4 cylinders was calculated from the collected data and average wasdetermined.
The average density of the 4 cylinders was 9.79 (g/cm^3). This value was compared with the density that wasretrieved from the graph which was 9.52 0.01g/cm^3. The difference between thesetwo values is not very significant but the values are not equal within therange of the calculated random errors. As mentioned before this could be due tomiscalculations and rounding the numbers.
Ininvestigation 2 we calculated the random error of radioactive emissions byregistering 20 entries during 20 minutes and inputted the data in table 1.2. Itwas stated that the equipment used to measure different quantities is notprecise but this is not the only factor that can affect random error.Therefore, we have to take other factors into consideration whenever we performmeasurements. From table 1.2 we calculated the average of counts/60s and it was20.8 counts/60s.
We compared this value to another group and theirs was 21.8 whichwas very similar to ours. From the histogram we calculated a random error ( of 3.39 counts/60swhich was different from the random error of the other group 4.25 counts/60s.This can be due to different bin sizes used in the histogram.
Next a standarddeviation ( )of 4.71 counts/60s wasfound from the data. The standard deviation was not similar to the random errorwe calculated they differed with a factor of 1.4. This could be due tomiscalculation as well and to not having a big enough data pool to have a moreprecise standard deviation.
To improvethese results the measurements of the mass, diameter, length could be takenmore carefully. The data pool could be increased in order to have morestatistically significant results. Also mistakes could have been madethroughout the steps, and could be fixed if you go back to double check.Questions1. When you tare the scale, it gets rid of the weight that wasthere before.
This would have changed our mass error it would have increasedit.2. Mass = 250g Diameter= 10cm L=?In this problem the mass and diameter of the object aregiven. Also, the density of this object would be the same as the averagedensity of the 4 cylinders from Investigation 1 since they are the samematerial. Therefore, using the density equation, we can solve for the Volume. After plugging in the values, volume (V) is equal to 25.54cm^3. Now we can proceed to find the length of the object by plugging in thevalues in volume equation and solving for L.
After plugging in the V and D values which are 25.54cm^3 and10 cm respectively we find the length (L) equal to 0.32cm.3.
Radius of sphere = 10 cmMass=? Thedensity of the sphere is the same as the one of the 4 cylinders fromInvestigation 1 since they are made of the same material, equal to 9.799g/cm^3. The volume of the sphere is found by the formula .
Volume is equal to 4,186.67 cm^3. The mass is found by thedensity formula p=m/V. Therefore, mass is equal to 41,025.
18g.4. In order to challenge the speed given by either thespeedometer or the radar gun, it could be argued that the speedometer was notcalibrated correctly which is a systematic error, or that the radar gunperformed a random error since no equipment gives the true measurement and isprecise.
5. If we would calculate the data from two Geiger counters thestandard deviation would decrease because the data pool would increase. AcknowledgementsFirstly, I would like to thank my TA, Laxmi Pandey, for being veryhelpful in explaining and walking us through the steps. Also, I would like tothank my lab partner, Chayanne Gumbs, for her full effort and being a good teamworker.