The goal of this
experiment was to find the errors associated with uncertainties during
measurement of the given objects. During this experiment,
the lengths and masses of objects were measured using the basic equipment. Based on the equipment used, the
precision of the measurement changes. However, the measurements cannot be exact
no matter how precise the equipment can be. In the first investigation, we
measured mass, diameter and length of 4 metal cylinders in order to calculate
their density and volume using the equations:

                                                                             
(1.1)

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                  (1.2)

 

The unit for mass is in (g), the density
in (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 of
mass vs volume in order to check for accuracy. In the second investigation, a Geiger
counter was used to measure the number of background radioactive counts present
in the lab. This was performed in order to practice error calculations. The
data was collected, analyzed and random errors were calculated for each
measurement and calculation.

 

Investigation 1

 

            The setup for this investigation consists of a digital scale
and graduated cylinder which was used to
record and compare the mass of 4 metal cylinders. The length and diameter of
each cylinder were measured with a ruler. All this equipment was placed on the working desk in order to perform each
step.

            Each cylinder was numbered from 1-4. They were placed on
the digital scale respectively after it was tared and the mass was recorded in
table 1.1. Error in mass ?m(g) was calculated on the second step of the
procedure. The error estimate was ½ of the smallest increment of the digital
scale, which was ½ of 0.1g equal to 0.05g. the error in mass ?m(g) was then
calculated 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 these
measurements. Again, the error estimate was ½ of the smallest increment on the
ruler. In our case, both the absolute
error in length (?L) and absolute error in diameter (?D) were ½ of 0.01cm equal
to 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 cylinders
and they were inputted in Table 1.1 in the appropriate rows. We proceeded in
calculating the volume (V) using the measurements of the diameter and length
and inputting them in the equation V=
(?D^2L)/4. We then proceeded in measuring the error of each volume that was
measured using the formula:

Afterwards, we used a graduated cylinder to record the volume of the
largest cylinder (#4) so we could compare it to the volume we calculated with
the formula in order to determine which volume was more precise and had less
error.Next, the density of each of the 4 cylinders was calculated as well as
the relative error. In order to calculate the density, the mass and the volume
calculated in the previous steps were inputted
in the equation (1.1). Average density (

) was calculated using
the equation:

 

 and the relative error of the average density (

 ) was calculated using the formula:         

Next, we proceeded to
build a graph with mass(g) in the y-axis and volume (cm^3) in the x-axis in
order 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 used
to calculate again the average density of the cylinders and the relative error
since 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- Measurement
and calculations for massg, diametercm, lengthcm, volumecm^3, densityg/cm^3
and their relative errors for Investigation 1

 

In order for the density
and volume to be calculated the mass(m), length (L) and diameter (D) of 4 metal
cylinders was measured. Each of these measurements were used to plug in
equations (1.1) and (1.2) and from these calculations the values of the density
and volume derived and were inputted in the table above.

 

 

Figure
1.2-
Plot of mass vs Volume. The error of
slope is equal to 0.01.

The density and error calculated with IPL from the
graph values was 9.52

0.01g/cm^3.
The graph portrayed in Figure 1.2 was used to calculate a more precise error
and density from the slope. The values that were retrieved from the graph were
used for comparison in order to see how accurate our calculations using the
values from the table were. The error bars are not portrayed in the graph
because the error was very small it was 0.01. In comparison to the data that
was achieved from the table the slope of the graph (density) the values were
not equal to the range of the random errors. This could be due to the rounding
up our calculations and slightly wrong mass(m) measurements.

Investigation
2

To set up this
investigation the GMC-200 box was connected to the power cable. One cable was
inserted in the Data input and then it was connected to the computer. The
counter was turned on by pressing the red button on the left side of the
device. To begin with, the GQ GMCounter
PRO software was started in order to collect the data that was needed. Then we
proceeded to set up the software by going to Setting in the top menu, pressed
the 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 and
select Restart Counter. This step made sure all previous data was erased. Next
the program was set to automatically record the number of counts for 1 hour. The
first 20 entries were inputted in table 1.2. To calculate random error of these
data a histogram was put together. The number of bars needed for the histogram
was 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

 

Table
1.2-
Measurements of counts/60s for
Investigation 2. (Data sorted by time)

The data from the table was used to calculate the
average value of background counts per minute (n). This was done by taking the
average of all the values in the counts/60s column. The n value was equal to 20.8 counts/60s. This value was compared
with the value of another group and both values were similar but not equal.

Figure
1.3
– Number of background counts per minute.

The Histogram was used to estimate the error of the
average count. This was calculated by the
formula:

W
in this equation stands for the width that exists between the blocks that are ½
of the height of the tallest block in the histogram and in our case ½ of the
tallest block was the block with frequency 3. The error for the average count
was equal to 3.39 counts/60s.

Conclusion

      Experiment
1 consisted of two investigations in which random errors of measurements were
calculated. We used different methods to compare the accuracy of each one of
them in finding the value of density or volume. We also saw how every
measurement is not very precise and we always have to take into account many
factors.

      In
investigation 1 the mass, diameter and length were measured and their random
errors. From these measurements the density and volume were calculated. Firstly,
we calculated the value of the volume for each cylinder. The volume of the
largest cylinder (#4) was calculated via the equation but also with a graduated
cylinder. The volumes were slightly different. The volume we recorded from our
calculations was 7.69

1.23E-07 cm^3 and the volume from the graduated cylinder was 9.0

 0.05 cm^3. This change
could be due to miscalculations and wrong measurements. Next, the density of
the 4 cylinders was calculated from the collected data and average was
determined. The average density of the 4 cylinders was 9.79

(g/cm^3). This value was compared with the density that was
retrieved from the graph which was 9.52

0.01g/cm^3. The difference between these
two values is not very significant but the values are not equal within the
range of the calculated random errors. As mentioned before this could be due to
miscalculations and rounding the numbers.

      In
investigation 2 we calculated the random error of radioactive emissions by
registering 20 entries during 20 minutes and inputted the data in table 1.2. It
was stated that the equipment used to measure different quantities is not
precise but this is not the only factor that can affect random error.
Therefore, we have to take other factors into consideration whenever we perform
measurements. From table 1.2 we calculated the average of counts/60s and it was
20.8 counts/60s. We compared this value to another group and theirs was 21.8 which
was very similar to ours. From the histogram we calculated a random error (

 of 3.39 counts/60s
which 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 standard
deviation (

 )of 4.71 counts/60s was
found from the data. The standard deviation was not similar to the random error
we calculated they differed with a factor of 1.4. This could be due to
miscalculation as well and to not having a big enough data pool to have a more
precise standard deviation.

      To improve
these results the measurements of the mass, diameter, length could be taken
more carefully. The data pool could be increased in order to have more
statistically significant results. Also mistakes could have been made
throughout the steps, and could be fixed if you go back to double check.

Questions

1.    
When you tare the scale, it gets rid of the weight that was
there before. This would have changed our mass error it would have increased
it.

2.    
Mass = 250g 

Diameter= 10cm

L=?

In this problem the mass and diameter of the object are
given. Also, the density of this object would be the same as the average
density of the 4 cylinders from Investigation 1 since they are the same
material. Therefore, using the density equation, we can solve for the Volume.

 

 

After plugging in the values, volume (V) is equal to 25.54
cm^3. Now we can proceed to find the length of the object by plugging in the
values in volume equation and solving for L.

 

 

After plugging in the V and D values which are 25.54cm^3 and
10 cm respectively we find the length (L) equal to 0.32cm.

3.    
Radius of sphere = 10 cm

Mass=?

The
density of the sphere is the same as the one of the 4 cylinders from
Investigation 1 since they are made of the same material, equal to 9.799
g/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 the
density formula p=m/V. Therefore, mass is equal to 41,025.18g.

4.    
In order to challenge the speed given by either the
speedometer or the radar gun, it could be argued that the speedometer was not
calibrated correctly which is a systematic error, or that the radar gun
performed a random error since no equipment gives the true measurement and is
precise.

5.    
If we would calculate the data from two Geiger counters the
standard deviation would decrease because the data pool would increase.

 

Acknowledgements

Firstly, I would like to thank my TA, Laxmi Pandey, for being very
helpful in explaining and walking us through the steps. Also, I would like to
thank my lab partner, Chayanne Gumbs, for her full effort and being a good team
worker. 

 

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