Lab Report 1: Basic Measurement
Sean Michelino Lim (2022360012)
Department of Mechanical Engineering, Sampoerna University
MECH3316 - A: Instrumentation Laboratory
Nikolas Krisma Hadi Fernandez
March 6th, 2025
1
Table of Contents
Table of Contents ....................................................................................................................... 2
List of Figures ............................................................................................................................ 3
List of Tables ............................................................................................................................. 3
List of Equations ........................................................................................................................ 3
Chapter I Introduction ................................................................................................................ 4
Chapter II Methodology ............................................................................................................. 5
2.1 Materials and Apparatus used .......................................................................................... 5
2.2 Experimental Procedure ................................................................................................... 5
2.3 Key Equations .................................................................................................................. 6
Chapter III Result and Discussion ............................................................................................. 9
3.1 Result................................................................................................................................ 9
3.2 Discussion and analysis.................................................................................................. 14
Chapter IV Conclusion ............................................................................................................ 16
References ................................................................................................................................ 17
2
List of Figures
No table of figures entries found.
List of Tables
Table 1: Materials and Apparatus list ......................................................................................... 5
Table 2: Experiment 1 results .................................................................................................... 9
Table 3: Experiment 2 - block 1 ................................................................................................. 9
Table 4:Experiment 2 - block 2 ................................................................................................ 10
Table 5:Experiment 2 - block 3 ................................................................................................ 11
Table 6: Experiment 2 - block 4 ............................................................................................... 11
Table 7: Experiment 2 - Marble ............................................................................................... 12
Table 8: Experiment 2 – Metal ball.......................................................................................... 13
List of Equations
Equation 1: Shows the equation to calculate the least count of an analogue device. ................ 6
Equation 2: Shows the equation to calculate the least count of a Digital device....................... 6
Equation 3: Shows the equation for sample standard deviation ................................................ 6
Equation 4: Shows the equation for mean standard error .......................................................... 7
Equation 5: Shows the equation for expanded uncertainty (with 95% confidence) .................. 7
Equation 6: Shows the equation for Volume of a cube .............................................................. 7
Equation 7: Shows the equation for uncertainty of a cube’s volume ......................................... 7
Equation 8: Shows the equation for density .............................................................................. 7
Equation 9: Shows the equation for uncertainty of density ....................................................... 7
Equation 10: Shows the equation for Volume of a sphere ......................................................... 7
Equation 11: Shows the equation for uncertainty of a sphere’s volume .................................... 7
3
Chapter I Introduction
Measurement serves as a fundamental pillar of scientific inquiry and engineering
practice, playing a crucial role in quantifying, analysing, and understanding the physical world.
From basic physics to complex engineering applications, precise and accurate measurements
are indispensable for validating theoretical models and ensuring the reliability of experimental
results. This laboratory module represents an essential starting point in the Instrumentation
Laboratory course, focusing on the foundational principles of measurement techniques and
uncertainty analysis.
One of the primary objectives of this experiment is to explore the importance of
repeated measurements, which not only enhance the reliability of collected data but also
provide a means to assess the precision and accuracy of different measuring instruments.
Measurement errors are inherent in any experimental procedure due to factors such as
instrument limitations, human perception, and environmental conditions. Thus, understanding
the concepts of uncertainty, resolution, and precision is vital in ensuring that reported values
are both meaningful and statistically robust.
This experiment specifically focuses on dimensional measurements, utilizing two
widely used precision instruments: the vernier calliper and the micrometre. These tools allow
for highly accurate readings by improving upon the resolution limits of simpler measuring
devices like standard rulers. By measuring the dimensions of a three-dimensional object
multiple times, students will gain hands-on experience in data acquisition, error reduction
techniques, and uncertainty quantification. Additionally, this lab introduces the principles of
significant figures and propagation of uncertainty, both of which are critical in engineering
measurements.
Through this experiment, students will not only develop a deeper understanding of
measurement fundamentals but also cultivate essential analytical skills necessary for future
experimental and industrial applications. The ability to critically evaluate measurement data,
identify sources of error, and apply statistical methods to improve accuracy forms the backbone
of effective engineering practice.
4
Chapter II Methodology
2.1 Materials and Apparatus used
Table 1: Materials and Apparatus list
Materials
Name
Amount
Note
Blocks
4x
4 blocks made of varying materials are provided
Balls
2x
2 balls made from different materials are provided
Apparatus
Digital Multimeter
1x
Stopwatch
1x
Digital weight
scale
1x
Triple arm balance
1x
Vernier calliper
1x
Micrometre
1x
Thermometer
1x
Thermogun
1x
Ruler
1x
Includes:
Amperemeter
Voltmeter
ohmmeter
Table 1 displays the materials and apparatus that are necessary and crucial to conduct
this experiment including the names and quantity and a note for detail which is required to
replicate this experiment
2.2 Experimental Procedure
This experiment follows the following procedure:
Setup:
5
•
Prepare all the apparatus in a table in a line to simplify the experiment in the any order
with vernier calliper and micrometre at the end, followed by the materials
First experiment:
•
•
Identify the least count of each apparatus by looking at either the digital or analogue
scale
o
For digital, we use the smallest digit displayed as our least count as shown in
Equation 2
o
For Analogue we use the smallest digit divided by 2 as we can still determine
in-between the lines for an analogue scale as seen in Equation 1
o
Repeat for all apparatus provided
Input the data obtained into the lab module table
Second experiment:
•
Take the provided vernier calliper and micrometer
•
Calibrate the micrometer using the given tool
•
Calculate the dimensions of the materials (blocks and balls)
•
o
For the blocks, measure 3 sides 6 times each using the vernier callipers
o
For the balls, measure the diameter of the ball 6 times.
Input the obtained data on the lab module table
2.3 Key Equations
The following equations are essential within this experiment, to analyse the resulting
data and obtain the needed results:
1
(1)
𝛥𝑥 = 𝐿𝐶
2
Equation 1: Shows the equation to calculate the least count of an analogue device.
(2)
𝛥𝑥 = 𝐿𝐶
Equation 2: Shows the equation to calculate the least count of a Digital device.
𝑆𝑥 = √∑𝑁
𝑖=1
(𝑥𝑖 −𝑥̅ )2
𝑁−1
(3)
Equation 3: Shows the equation for sample standard deviation
𝑆𝑥̅ =
𝑆𝑥
√𝑁
(4)
6
Equation 4: Shows the equation for mean standard error
(5)
𝑢𝑥̅ = 2𝑡𝑣,95 𝑆𝑥̅
Equation 5: Shows the equation for expanded uncertainty (with 95% confidence)
(6)
𝑉 =𝐿∗𝑊∗𝐻
Equation 6: Shows the equation for Volume of a cube
𝑢
2
𝑢
2
𝑢
2
𝑢𝑉 = 𝑉 ∗ √( 𝐿 ) + ( 𝑊) + ( 𝐻 )
𝐿
𝑊
𝐻
(7)
Equation 7: Shows the equation for uncertainty of a cube’s volume [1]
𝜌=
𝑚
(8)
𝑉
Equation 8: Shows the equation for density
2
𝑢
𝑢
2
𝑢𝜌 = 𝜌 ∗ √( 𝑚 ) + ( 𝑉 )
𝑚
𝑉
(9)
Equation 9: Shows the equation for uncertainty of density [1]
4
𝑉 = 𝜋𝑟 3
(10)
3
Equation 10: Shows the equation for Volume of a sphere
𝑢
𝑢𝑉 = 𝑉 ∗ 3 ∗ ( 𝑟 )
𝑟
(11)
Equation 11: Shows the equation for uncertainty of a sphere’s volume [1]
𝛿=
𝑢𝑥̅
𝑥̅
(12)
Equation 12: : Shows the equation for relative uncertainty
Key terms:
Δx: Measurement uncertainty
LC: Least count of the measuring instrument
𝑆𝑥 : Sample standard deviation
𝑥𝑖 : Individual measurement
𝑥̅ : Mean (average) of the measurements
7
N: Number of measurements taken
𝑆𝑥̅ : Standard uncertainty of the mean (Standard Error)
𝑢𝑥̅ : Uncertainty
𝑡𝑣,95 : t-distribution critical value for 95% confidence level
𝑣: Degrees of freedom
V: Volume
L: Length
W: Width
H: Height
Ρ; density
m: mass
𝛿 : Relative uncertainty
8
Chapter III Result and Discussion
3.1 Result
Table 2: Experiment 1 results
Name of Device
Least count
Vernier Calliper
0.01 mm
Ruler
0.25 mm
Thermometer
0.05°c
Thermo gun
0.1°c
Triple Arm Balance
0.0005 g
Voltmeter
0.1 mV
Amperemeter
0.1 μA
Stopwatch
0.01 s
Digital Weight Scale
1g
Micrometre
0.0005 mm
Table 2 shows the data obtained from observation on the least count (LC) of each
apparatus as per experiment 1. Using Equation 1 and 2, which dictates that a digital scale will
have an LC equivalent to their smallest digit increment whilst an analogue will have LC of half
of their smallest increment.
Table 3: Experiment 2 - block 1
Block 1
Mass: 69.9±0.0005 gr
Data #
1
Length (mm)
20
Width (mm)
19.96
Height (mm)
20.06
9
2
20
19.98
20.04
3
20
20
20
4
20
20
20.06
5
20
20
20.04
6
20
20.02
20.04
𝑆𝑥
0
0.021
0.022
𝑆𝑥̅
0
0.008
0.009
𝑢𝑥̅
0
0.017
0.018
𝑥̅
Length: 20
Width: 19.993
Height: 20.04
Table 4:Experiment 2 - block 2
Block 2
Mass: 22.679±0.0005 gr
Data #
Length (mm)
Width (mm)
Height (mm)
1
19.86
19.98
19.94
2
19.98
20
19.92
3
20.02
20.1
20
4
19.94
20.04
20.02
5
19.96
20.06
19.96
6
19.98
19.98
20
10
𝑆𝑥
0.054
0.048
0.039
𝑆𝑥̅
0.022
0.02
0.016
𝑢𝑥̅
0.044
0.04
0.032
𝑥̅
Length: 19.957
Width: 20.027
Height: 19.973
Table 5:Experiment 2 - block 3
Block 3
Mass: 65.8±0.0005 gr
Data #
Length (mm)
Width (mm)
Height (mm)
1
20.02
20.2
19.86
2
20
20.1
19.92
3
20.2
20.2
19.82
4
20.1
20
19.9
5
20.2
20
19.8
6
20.1
20.06
19.82
𝑆𝑥
0.085
0.091
0.048
𝑆𝑥̅
0.035
0.037
0.02
𝑢𝑥̅
0.07
0.074
0.04
𝑥̅
Length: 20.103
Width: 20.093
Height: 19.853
Table 6: Experiment 2 - block 4
11
Block 4
Mass: 60.39±0.0005 gr
Data #
Length (mm)
Width (mm)
Height (mm)
1
19.9
19.82
20.1
2
19.92
20.2
20.1
3
19.88
19.96
20.04
4
19.9
19.9
20.01
5
19.92
19.9
20
6
19.92
19.88
20.04
𝑆𝑥
0.016
0.134
0.043
𝑆𝑥̅
0.007
0.055
0.018
𝑢𝑥̅
0.013
0.109
0.035
𝑥̅
Length: 19.907
Width: 19.943
Height: 20.048
Table 3,4,5,6 shows the data obtained from experiment 2 regarding measurement of box
and balls. Using equations 3,4, and 5 we can complete the table in regards to its standard
deviation, mean standard deviation, and the uncertainty. Adding the last data of mean, we have
completed the data for tables 3, 4,5,6 which will also contain the same exact information on all
of the different blocks.
Table 7: Experiment 2 - Marble
Marble
Mass: 4.82±0.0005 gr
Data #
1
Diameter (mm)
Radius (mm)
15.03
7.515
12
2
15.11
7.555
3
15.125
7.5625
4
15.42
7.71
5
15.46
7.73
6
15.05
7.525
𝑆𝑥
0.19
0.095
𝑆𝑥̅
0.078
0.039
𝑢𝑥̅
0.155
0.078
𝑥̅
Diameter: 15.199
Radius: 7.599
Table 8: Experiment 2 – Metal ball
Metal
Mass: 4.095±0.0005 gr
Data #
Diameter (mm)
Radius (mm)
1
10
5.00
2
10.01
5.005
3
10.005
5.0025
4
10.005
5.0025
5
10.005
5.0025
6
10.005
5.0025
13
𝑆𝑥
0.003
0.002
𝑆𝑥̅
0.001
0.001
𝑢𝑥̅
0.003
0.001
𝑥̅
Diameter: 10.005
Radius: 5.002
3.2 Discussion and analysis
Experiment 1
In measurement, several key terms define the quality and reliability of data [2]:
•
Precision refers to how consistently repeated measurements align with each other.
•
Accuracy describes how close a measurement is to the true value.
•
Sensitivity indicates how responsive an instrument is to small changes in the quantity
being measured.
•
Resolution defines the smallest detectable change an instrument can measure.
•
Error is the difference between the measured value and the actual value.
Measurement instruments used in this experiment can be categorized into five main
types based on what they measure: Length, Weight, Temperature, Time, Voltage, and Current
(Ampere).
Focusing on length measurement, we evaluated three instruments: the Vernier caliper,
Ruler, and Micrometer. Each varies in precision, with the micrometer being the most precise.
It offers an accuracy of ±0.0005 mm due to its high resolution of 0.001 mm, with it being an
analog device, Using Equation 1, we can determine that the micrometer allows for more precise
measurements, making it the most reliable instrument among the three for fine measurements.
Experiment 2
Using equations Equation 6, 7, 8, and 9 we can calculate both the density and volume
of the block whilst propagating the uncertainty generated by each of the measurement taken.
Block 1
Volume = 8.015±0.01 cm³
Density = 8.723±0.011 g/cm³
Block 2
Volume = 7.983±0.027 cm³
14
Density = 2.841±0.01 g/cm³
Block 3
Volume = 8.020±0.044 cm³
Density = 8.205±0.045 g/cm³
Block 4
Volume = 7.959±0.046 cm³
Density = 7.587±0.044 g/cm³
Using equations Equation 8, 9, 10, and 11 we can calculate both the density and volume of
the spherical materials whilst propagating the uncertainty generated by each of the
measurement taken.
Marble Ball
Volume = 1.838±0.056 cm³
Density = 2.622±0.08 g/cm³
Metal Ball
Volume = 0.524±0.0004 cm³
Density = 7.809±0.006 g/cm³
Using the density data obtained it is possible to determine the materials that make up
each of these 3D objects. Block 1 with 8.721±0.007 g/cm³ is close in density to nickel which
has a density of 8.6 g/cm³ [3]. Following that would be Block 2 with density of 2.841±0.009
g/cm³ which is indicative of aluminium with a density of 2.7 g/cm³ [4]. Block 3 with
8.205±0.036 g/cm³ is similar to brass at 8.4 g/cm³ [5]. Lastly Block 4 with 7.587±0.029 g/cm³
indicates stainless steel at 7.5 g/cm³ [6].
Following the blocks, the materials of the balls are also identifiable with marble ball at
2.622±0.043 g/cm material being glass at 2.8 g/cm [7]. Whilst the metal ball with a density of
7.81±0.01 g/cm³ indicates mild steel at 7.85 g/cm³ [8].
15
Chapter IV Conclusion
This experiment demonstrated the fundamental principles of measurement and
uncertainty analysis, emphasizing the importance of precision in experimental data. Through
repeated measurements and statistical methods, the volume and density of various solid objects
were determined with quantified uncertainties. The application of error propagation formulas
ensured that the reported values were statistically meaningful. By comparing the calculated
densities with reference materials, the composition of each object was approximated,
identifying materials such as nickel, aluminium, brass, stainless steel, glass, and mild steel. The
study highlighted the significance of accurate measurement techniques and proper uncertainty
analysis in engineering applications, reinforcing the necessity of rigorous data validation in
experimental procedures. Future improvements could include refining measurement tools and
minimizing systematic errors to enhance accuracy.
16
References
[1] "Uncertainty propagation," Volard Blog, Nov. 1, 2016. [Online]. Available:
https://volard.wordpress.com/2016/11/01/uncertainty-propagation/. [Accessed: Mar. 6,
2025].
[2] "Basic Measurement and Uncertainty," ENGR 3311 – Instrumentation Laboratory,
Experiment Module #01.
[3] Nickel Institute, “Properties of Nickel,” Accessed: Mar. 6, 2025. [Online]. Available:
https://nickelinstitute.org/en/nickel-applications/properties-of-nickel
[4] Thyssenkrupp Materials, “Density of Aluminium,” Accessed: Mar. 6, 2025. [Online].
Available: https://www.thyssenkrupp-materials.co.uk/density-of-aluminium.html
[5] KDM Fab, “Brass Density,” Accessed: Mar. 6, 2025. [Online]. Available:
https://kdmfab.com/brass-density
[6] Engineers Edge, “Densities of Metals and Elements Table,” Accessed: Mar. 6, 2025.
[Online].
Available:
https://www.engineersedge.com/materials/densities_of_metals_and_elements_table_1
3976.htm
[7] LibreTexts, “Glass Density Evidence,” Accessed: Mar. 6, 2025. [Online]. Available:
https://chem.libretexts.org/Ancillary_Materials/Exemplars_and_Case_Studies/Exempl
ars/Forensics/Glass_Density_Evidence
[8] Solitaire Overseas, “Density of Mild Steel,” Accessed: Mar. 6, 2025. [Online]. Available:
https://www.solitaire-overseas.com/blog/density-of-mildsteel/#:~:text=The%20current%20density%20of%20mild%20steel%20is%207.85%2
0g%2Fcm³,What%20is%20a%20mild%20steel%3F
17