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)

(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.

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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.