MARMARA UNIVERSITY
FACULTY OF
ENGINEERING PHYS 1103.2
PHYSICS LABORATORY
Section Number: 2
Group Number: 13
Submitted to: Funda
CIRIK AÇIKKAYA
Due Date: 22.10.2025
Dep
t
Student Id
Name Surname
Contribution
1
KM
M
150724851
Erdem AKYAZI
Measurements ,Results,Upload
Report,Theory
2
KM
M
150725033
Ahmed KIRAY
Mesaurements,Results
3
KM
M
150724824
Fatih ACAR
Conclusion, Results
1
Table of Contents
1 OBJECTIVES
2 THEORY
3 EQUIPMENT LIST
CALCULATIONS AND RESULTS
4.1. Part 1: Rectangular Prism
4.2. Part 2: Cylinder with Holes
5DISCUSSION AND CONCLUSION
6REFERENCES
7APPENDIX
2
OBJECTIVES
The objective of this experiment is to learn the correct use of basic measurement instruments
such as a ruler and a caliper. Within the scope of the experiment, the external dimensions of
given regular geometric objects (a rectangular prism and a cylinder with holes) were
measured, and their volumes were calculated from these measurements. The accuracy and
error margins between measurements taken with the ruler and the caliper were compared, and
the densities of the objects were determined using different measurement techniques.
THEORY
Systematic Errors
Systematic Errors are errors that are consistent and repeatable. They typically arise from
faulty or improperly calibrated equipment, or from a flawed experimental method. For
example, a balance that consistently reads 0.5 g higher than the true mass would introduce a
systematic error. These errors affect the accuracy of a measurement, causing the results to be
consistently skewed in one direction.
Random Errors
Random Errors are unpredictable variations in the measured values. They can be caused by
uncontrollable fluctuations in experimental conditions or by the limits of the observer's
ability to read an instrument. For instance, estimating the last digit when reading a ruler can
introduce random errors. These errors affect the precision of a measurement, causing the
results to be scattered around an average value. Taking multiple measurements and
calculating their average can help minimize the effect of random errors.
Significant Figures
The precision of a measurement is properly communicated by its number of significant
figures. Significant figures in a measurement include all the digits that are known with
certainty plus one final digit that is estimated. When performing calculations with measured
values, the result cannot be more precise than the least precise measurement used. Specific
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rules for addition/subtraction and multiplication/division must be followed to ensure the final
result has the correct number of significant figures.
Accuracy and Precision
Experimental errors are the difference between a measurement and the true value or between
two measured values. Under the effect of these errors, the reliability of the measured values
are analyzed by its accuracy and precision. Accuracy and precision are two important factors
to consider when taking data from measurements. The accuracy of an experiment is a measure
of how close the experimental result comes to the true (or accepted) value. It is related to a
measure of the correctness of the result. Source of systematic errors determines how accurate
the measurement is. The precision of an experiment is a measure of how reproducible the result
is. That is, it is a measure of the magnitude of the uncertainty of the result due to random errors.
It is related to a measure of repeatability of the result and therefore, it is usually limited by the
sources of the random errors.
Both accuracy and precision reflect how close a measurement is to an actual value, but
accuracy reflects how close a measurement is to a known or accepted value, while precision
reflects how reproducible measurements are, even if they are far from the accepted value
To clarify these definitions, consider an example about throwing and striking a bullet to a target
(Figure 3).
In the Figure 3, in the Part-(a), there is an example showing an
example of striking which is neither precise nor accurate. Because the aim of throwing the
bullets is to hit the target on its center, they are not in the middle of the inner circle and also
the differences between them are very large. Part-(b) shows an example of striking which is
accurate but not precise. As compared to the results in part-a, the bullets are near the center of
the target but they are not close to each other. Part-(c) shows an example of striking which is
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precise but not accurate. As seen in Part-(c), the bullets are in a group which means the results
are approximately the same; the difference between the bullets is very low. However, they are
not located near the center of the target. Lastly, Part-(d) shows an example of striking which is
both accurate and precise. This means that each of the trial gives the true (accepted) result, the
bullets are located near the center of the target, and also, they give the similar results, the
difference between the bullets is small.
The accuracy of an experiment often depends on both systematic and random errors. The
precision of an experiment typically depends on random errors. (Bkz.phys1103 manual_1)
Error Propagation
Addition and Subtraction
If a result R is obtained by adding or subtracting two measurements, x ± Δx and y ± Δy, the
absolute uncertainty in R is the sum of the absolute uncertainties.
ΔR = Δx + Δy
Multiplication and Division
If a result R is obtained by multiplying or dividing two measurements, the relative
uncertainty in R is the sum of the relative uncertainties of the individual measurements.
ΔR / |R| = Δx / |x| + Δy / |y|
General Formula for Error Propagation
A more general and widely applicable method for error propagation uses partial derivatives.
For a function f(x, y, z) that depends on several measured variables, the uncertainty in f,
denoted as Δf, is given by:
5
Δf = sqrt[(∂f/∂x Δx)² + (∂f/∂y Δy)² + (∂f/∂z Δz)²]
Example for Arithmetic Mean
This formula can be used to find the uncertainty in the arithmetic mean. For an average of
three measurements,
H = (H₁ + H₂ + H₃) / 3
the partial derivatives are
∂H/∂H₁ = 1/3, ∂H/∂H₂ = 1/3, ∂H/∂H₃ = 1/3
If we assume the uncertainty of each individual measurement is the same (ΔH₁ = ΔH₂ = ΔH₃
= Δh), the uncertainty in the mean (ΔH) becomes:
ΔH = sqrt[3 (Δh / 3)²] = Δh / sqrt(3)
This important result shows that for n measurements, the uncertainty of the mean is √n times
smaller than the uncertainty of a single measurement.
EQUIPMENT LIST
Rectangular Prismatic Object
Cylindric Object with Holes
Digital Balance (Precision: ± 0.005 g)
Caliper (Precision: ± 0.01 mm)
Ruler (Precision: ± 0.5 mm)
CALCULATIONS AND RESULTS
In this section, final values for volume and density were derived from the raw experimental
data. For each dimension, the arithmetic mean of three separate measurements was used as
the best estimate. The uncertainties in the final results were determined using the error
propagation formulas presented in the Theory section. The experiment was conducted in two
parts.
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Part 1: Rectangular Prism
Mass of the Prism (m): 122 ± 0.005 g
Table 1: Raw Measurements and Average Values for the Prism
Dimension
Instrument
Measureme Measureme Measureme Average
nt 1 (mm)
nt 2 (mm)
nt 3 (mm)
(mm)
a
Ruler
35.0
35.0
34.0
34.6
a
Caliper
35.1
35.1
35.0
35.06
b
Ruler
55.0
55.0
54.0
54.6
b
Caliper
55.2
55.1
54.9
55.06
c
Ruler
45.0
45.0
44.0
44.6
c
Caliper
45.1
45.1
45.0
45.06
The average values in Table 1 were calculated using Equation (1) from the Theory section.
Table 2:Volume of the Prism
Instrument
Average
Calculated
Uncertainty
Dimensions (a, Volume (cm³)
(ΔV) (cm³)
b, c) (cm)
Ruler
3.46, 5.46, 4.46 84.3
± 2.9
Caliper
3.506, 5.506,
86.98
± 0.06
4.506
The volume of the prism in Table 2 was calculated using the formula:
Final Volume
(V ± ΔV) (cm³)
84.3 ± 2.9
86.98 ± 0.06
V=a·b·c
Table 3: Density of the Prism
Instrument
Mass (g)
Volume
(cm³)
Calculated
Density
(g/cm³)
1.45
1.403
Uncertainty Final
(Δρ) (g/cm³) Density (ρ ±
Δρ) (g/cm³)
Ruler
122
84.3
0.05
1.45 ± 0.05
Caliper
122
86.98
0.001
1.403 ±
0.001
The density of the prism in Table 3 was calculated using the formula
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ρ=m/V
Part 2: Cylinder with Holes
Mass of the Cylinder (m): 134.13 ± 0.005 g
Table 4: Raw Measurements and Average Values for the Cylinder
Dimension
Instrument
D
Ruler
Caliper
Ruler
Caliper
Ruler
Caliper
Ruler
Caliper
Ruler
Caliper
Ruler
Caliper
H
d1
h1
d2
h2
Measurement
1 (mm)
44.0
45.0
59.0
57.0
13.0
13.2
21.0
21.0
9.0
10.0
12.0
12.0
Measurement
2 (mm)
45.0
45.2
58.0
59.9
13.0
12.0
20.0
19.3
9.0
8.0
12.0
12.0
Measurement
3 (mm)
44.0
45.1
60.0
60.0
13.0
12.0
21.0
21.0
9.0
8.1
12.0
12.0
Average
(mm)
44.33
45.10
59.00
58.97
13.00
12.40
20.67
20.43
9.00
8.70
12.00
12.00
Table 5: Volume of the Cylinder
Instrument
Calculated Net
Volume (cm³)
87.4
90.95
Ruler
Caliper
Uncertainty (ΔV)
(cm³)
3.2
0.07
Final Volume (V ±
ΔV) (cm³)
87.4 ± 3.2
90.95 ± 0.07
Table 6: Density of the Cylinder
Instrument
Mass (g)
Volume
(cm³)
Ruler
Caliper
134.13
134.13
87.4
90.95
Calculated
Density
(g/cm³)
1.53
1.475
Uncertainty
(Δρ) (g/cm³)
0.06
0.001
Final Density
(ρ ± Δρ)
(g/cm³)
1.53 ± 0.06
1.475 ± 0.001
8
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Conclusion
In this experiment, we aimed to measure the dimensions, volume, and density of two objects
(a rectangular prism and a cylinder with holes) using two different measuring instruments (a
ruler and a caliper). Through these measurements, we examined the concepts of accuracy,
precision, and error propagation in physics.
According to the results summarized in Tables 1–6, the measurements taken with the caliper
consistently showed smaller uncertainties compared to measuraments obtained with the ruler.
The propagated errors, calculated using the formulas given in the Theory section, confirmed
that the precision of the caliper is significantly higher. For example, the volume of the prism
measured by the caliper (Table 2) was 86.98±0.06 cm3, whereas the same volume measured
with the ruler had an uncertainty almost fifty times larger (84.3±2.9 cm3). This was also
observed for the cylinder (Table 5), indicating that the caliper provides both higher precision
and accuracy.
As discussed in the Theory section and illustrated in Figure 3, accuracy and precision are
different but related aspects of measurement quality. The data obtained in this experiment
support that systematic errors mostly affect accuracy, while random errors affect the
precision. The difference between the results of the two instruments shows how these error
types manifest in real measurements.
In conclusion, this experiment successfully demonstrated the relationship between
measurement accuracy, precision, and error propagation. It also showed the importance of
using appropriate tools for obtaining reliable physical data. Understanding and minimizing
measurement errors are very important skills in all experimental work, and this lab provided a
practical foundation for these concepts.
10
Referances

Physics 1103.2 First Lab Manual
Appendix
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