Sooknauth_Experiment1LabReport (4)
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Report for Experiment #1
Measurement
Diana Sooknauth
Lab Partner:
Pricilla De Guzman
TA:
Emily Tsai
January 23
rd
, 2023
Abstract
In this experiment, our overall goal was to learn how to use measurements and account
for errors in measurements. For investigation 1, we were trying to figure out what material the
four cylinders given to us were made of. We measured the mass and volume of the cylinders and
used that data to figure out the density. What we found was the average density of the cylinders
was 9.76
± .
775
g/ cm
3
and therefore the cylinders are most likely made out of brass. For the
second part of the experiment, we used a Geiger counter to measure the background radiation
count rate. We collected data and found that the average background radiation count rate was
19.5 ± 4 counts/60s.
Introduction
In this experiment, our first goal was to figure out the density of the cylinders given to us by
using mass and volume. Density is defined as the mass of a unit volume of a material substance
and can be found by dividing mass by volume. By finding the density of the material, we would
be able to identify it since the density of a material does not change significantly unless it is
facing a change in temperature or pressure. Our second goal was to figure out the average
background radiation count rate in our lab room. The particles around us naturally contain
radiation, which we call background radiation. The air around us contains unstable isotopes and
when measured, they are registered as radioactive decay. Radioactive decay occurs randomly and
therefore by collecting data, there are noticeable fluctuations in the measurements. The
background radiation was able to be measured by a Geiger counter. The Geiger counter has a
stable gas in a chamber of the device and when that gas is exposed to radioactive particles, like
the ones in the air around us, the gas ionizes and creates an electrical current that the counter
records. Our third goal was to learn how to calculate and account for different types of errors in
the data since every measurement has limited precision and this will be something we have to do
for the rest of our labs.
The objectives of this experiment were to get the length and weight measurements and
determine their uncertainties/ errors, use that data to obtain values for derived quantities,
understand how errors in data propagate, learn how to plot data, and use graphical methods for
data analysis, and understand basic aspects of statistics.
The different types of errors we had to measure and consider with this experiment are
instrumental, systematic, and random. Instrumental errors occur when the instruments and
equipment being used are inaccurate. Systematic errors are experimental flaws that cause
measurements to be consistently inaccurate in the same way each time they are taken. These
could occur due to limitations of the equipment or procedure. Random errors are unpredictable
statistical fluctuations; they occur due to chance.
Because this experiment involved using data to make calculations, the use of standard
deviation and the standard error of the mean were needed. The standard deviation of a data set is
a measurement of the amount of variability that the individual data values have compared to the
mean, which is the average value of a data set. The standard error of the mean is how far the
sample mean of the data is likely to be from the population mean, meaning the average of the
rest of the data. The computed standard deviation for investigation 2 in this experiment was 4.00
counts/60s. The computed standard error in the mean was 0.517 counts/60s.
Investigation 1
For investigation 1, the equipment needed was four metal cylinders of the same material, a
ruler, a digital scale, and a 100 ml graduated cylinder. The ruler was used to measure the length
and diameter of each cylinder, the digital scale was used to weigh each cylinder, and the
graduated cylinder was used as another way to figure out the volume of each cylinder.
To begin our investigation, we had to collect data for mass, density, and length as well as
their errors and relative errors. To collect the data for mass, we used a digital scale. We made
sure the scale was zeroed before we began the measurement. We started measuring each cylinder
and recorded their respective masses in grams. We kept track of the cylinders and their data by
numbering them 1-4. Cylinder 1 had a mass of 68.7 g. Cylinder 2 had a mass of 28.7g. Cylinder
3 had a mass of 34.6g. Cylinder 4 had a mass of 8.6g. After documenting the cylinders’ mass, we
had to calculate the error in mass, which is represented by δm. To do so we just used the fact that
a good estimate of the error is half of the smallest increment you are measuring. In this case, the
error in mass was determined to be 0.050 grams for each cylinder, since .100 g would be the
smallest increment when using grams. We then had to find the relative errors in mass. In order to
find the relative errors in mass, represented by δm/m, we divided each cylinder’s respective error
in mass by their mass. Cylinder 1 had a relative error in mass of 0.001. Cylinder 2 had a relative
error in mass of 0.002. Cylinder 3 had a relative error in mass of 0.001. Cylinder 4 had a relative
error in mass of 0.006.
Once we finished collecting data for mass, we now had to collect data for the diameter
and length of each cylinder. We used a ruler to help us collect data for this. We measured each
cylinder’s diameter with the ruler. Cylinders 1, 2, 3, and 4 had a diameter of 1.20 cm, 0.900 cm,
0.900 cm, and 0.600 cm respectively. Similar to finding mass, we had to find the error in
diameter (
δD)
which we also found by taking half of the smallest increment of a centimeter. This
gave us an error of .050 centimeters for each cylinder. We then had to find the relative error as
we did with mass.
The relative error in the diameter of Cylinder 1 was 0.042. The relative error
in the diameter of Cylinder 2 was 0.056. The relative error in the diameter of Cylinder 3 was
0.056. The relative error in the diameter of Cylinder 4 was 0.038.
After finding the diameter of each cylinder, we measured the length of each cylinder with
the ruler. Cylinders 1,2,3, and 4 had lengths of 6.20 cm, 4.60 cm, 5.50 cm, and 3.00 cm
respectively. The error in length for each cylinder (δL) was 0.050 cm, which makes sense as we
used centimeters for the diameter and had the same error. The relative error in length (δL/L) for
Cylinder 1 was 0.008. For Cylinder 2 it was 0.011. For Cylinder 3 it was 0.009. For Cylinder 4 it
was 0.017. We now had all the data needed to move forward.
We now had to find the volume of each cylinder. The equation used to find the volume of
each cylinder with the data of the diameter and length we had was:
V
=
π D
2
L
4
. For Cylinder
1, we calculated a volume of 7.01 cm
3
. Cylinder 2 had a volume of 2.92 cm
3
. Cylinder 3 had a
volume of 3.50 cm
3
. Cylinder 4 had a volume of 0.848
cm
3
. After finding this data, we then had
to calculate the relative error for volume. The equation given to us to find the relative error in
volume was:
δL
L
¿
2
2
δD
D
¿
2
+
¿
¿
δV
V
=
√
¿
(1)
The relative error in volume for Cylinders 1, 2, 3, and 4 were 0.084, 0.112, 0.111, and 0.167
respectively. Unlike with the other measurements, we found the relative error first before finding
the error. We used this equation to find the relative error of the volume, but it was also useful to
find the error in volume. To find the error in volume we simply multiply the cylinder’s individual
volumes by their respective relative errors. By doing this, we found that the error in volume for
Cylinder 1 was 0.587 cm
3
. For Cylinder 2 it was 0.327 cm
3
. For Cylinder 3 it was 0.390 cm
3
. For
Cylinder 4 it was 0.142 cm
3
.
We then conducted another test to find the volume for Cylinder 1. We took the graduated
cylinder and put in 60 cm
3
of water. When we placed Cylinder 1 inside, the water rose up to
about 70 cm
3
. The displacement of the water was about 10 cm
3
. Calculating the displacement of
the water indicates volume, so we knew this meant that the volume given to us by the graduated
cylinder was about 10 cm
3
. Compared to the result we got when using the equation to find
volume, which was 7.01 cm
3
we believe that the results we got when using the volume equation
is more precise. We believe this because there is a difference of almost 3 cm
3
between the two.
The graduated cylinder is not as reliable because sometimes it could be hard to read the
measurements on it and get an exact measurement. We were not able to get an exact
measurement when using the graduated cylinder, which is why we had to estimate that the
displacement was about 10 cm
3
.
After conducting the test with the graduated cylinder, we then had to start calculating the
average density of the cylinders, which was our actual goal for this investigation. To find the
average density, we first had to find the density (
ρ
¿
of each individual cylinder. In order to do
so we used the equation:
ρ
=
m
V
, The densities found for Cylinder 1, 2, 3 and 4 were 9.80 g/
cm
3
, 9.81 g/ cm
3
, 9.89 g/ cm
3
, and 10.1 g/ cm
3
respectively. In order to find the relative error in
density, the equation below was used:
δ V
V
¿
2
δ m
m
¿
2
+
¿
¿
δ ρ
ρ
=
√
¿
(2)
The relative densities for Cylinder 1, 2, 3 and 4 were 0.084, 0.112, 0.111, and 0.168 respectively.
Similar to how we found the error in volume, we were also able to find the error in density (δρ).
The error in densities for Cylinder 1, 2, 3 and 4 were 0.821 g/ cm
3
, 1.10 g/ cm
3
, 1.10 g/ cm
3
, and
1.70 g/ cm
3
respectively. All of the calculated data can be seen in Table 1 below. To find the
average density and error in density, we added the densities and divided them by four; we did the
same thing for the error in densities. The average density was 9.91 g/ cm
3
and the average error in
density was 1.18 g/ cm
3
.
Cylinder
1
2
3
4
m(g)
68.7
28.7
34.6
8.6
δm (g)
0.050
0.050
0.050
0.050
δm/m
0.001
0.002
0.001
0.006
L (cm)
6.20
4.60
5.50
3.00
δL (cm)
0.050
0.050
0.050
0.050
δL/L
0.008
0.011
0.009
0.017
D (cm)
1.20
0.900
0.900
0.600
δD (cm)
0.050
0.050
0.050
0.050
δD/D
0.042
0.056
0.056
0.083
V (cm3)
7.01
2.92
3.50
0.848
δV (cm3)
0.587
0.327
0.390
0.142
δV/V
0.084
0.112
0.111
0.167
ρ(g/cm3)
9.80
9.81
9.89
10.1
δρ(g/cm3)
0.821
1.10
1.10
1.70
δρ/ρ
0.084
0.112
0.111
0.168
Table 1:
Data collected for investigation 1.
After we collected our numerical data, it was time to express our data graphically to obtain
the best value of density. Using the mass data for the y-axis and the volume data for the x-axis,
Graph 1 was plotted:
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