Introduction to mass principle

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Papua New Guinea University of Technology
Department of Mechanical Engineering
ME 311 – Experimental Engineering
ME 261 – Statics
Experiment 1: Introduction to mass principle.
Title: Introduction to Mass principle
Abstract
Every physical object whether a living thing or non-living thing, from a crawling ant to a massive stone
has what we called mass. Mass can be defined as the amount of matter intrinsic to a body and the
numerical measure of inertia. Mass is both the property of a physical body and a measure of its
resistance to acceleration (a change in its state of motion) when a net force is applied. The objects mass
also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is
the kilogram (kg) which is in metric system and in British Imperial system of unit is the pound (lb.). The
mass of an object can be determine using scale, spring balance, etc. Also it can be determined by using
the density formula. Mass is used to determine the density of an objects.
An important property of any material is its density, defined as its mass per unit volume. A homogenous
material such as ice or iron has the same density throughout. We use the Greek letter  (rho) for
density. If the mass m of homogeneous material has volume V, the density  is  = / .
The density of some materials varies from point to point within the material such as earth atmosphere
(which is less dense at high altitude) and oceans (which are denser at greater depth). In general, the
density of a material depends on environmental factors such as temperature and pressure. The SI unit
for density is the kilogram per cubic meter (1kg/mᵌ).
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Table of content
Contents
Introduction .................................................................................................................................................. 3
Theory ........................................................................................................................................................... 3
Experimental apparatus ................................................................................................................................ 3
Test procedure .............................................................................................................................................. 4
Results ........................................................................................................................................................... 4
Sample Calculation:................................................................................................................................... 5
Discussion of Results ..................................................................................................................................... 6
Summary ....................................................................................................................................................... 7
Bibliography .................................................................................................................................................. 8
Appendices.................................................................................................................................................... 8
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Introduction
It is known from experience that a heavier body has a great deal of inertia and a light body has very
little. The concept of mass is used as a measurement of inertia
Mass = amount of matter in a body
The measurement of mass is by comparison with the standard kilogram (kg) of mass, a carefully prepare
platinum cylinder that was about 2.2 lbs. The gram mass commonly used is 10-3 kg. In the S.I System of
unit mass is measured in kilograms. Mass is proportional, but not equal to the local gravitational
acceleration.
Theory
In most laboratories, mass is usually measured using a spring balance. A spring balance measures the
weight of an object by opposing the force of gravity acting with the force of an extended spring. It works
by Hooke's Law, which states that the force needed to extend a spring is proportional to the distance that
spring is extended from its rest position. Therefore, the scale markings on the spring balance are equally
spaced.
Another method of measuring mass, especially for the irregular shaped objects, is by measuring the volume
of the water displaced by the object and calculating using the formula:

→  = 

Where ρ is the density of water, m is the mass to be calculated and V is the volume of water displaced
by the object.
=
The third method is for the regular shaped objects where length, breadth, height, etc. are measured
using ruler, micrometer, etc., the volume is calculated and then mass is calculated using the above
formula where density is the density of the object to be measured.
Experimental apparatus



-
Spring Balance
Steel Rule
Four (4) Specimen
Specimen 1: Small steel cylinder
Specimen 2: Medium Steel cylinder
Specimen 3: Large Specimen cylinder
Specimen 4: Aluminium Cylinder
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Test procedure
1. The spring balance was set to zero
2. Each specimen was weighed three times and the results were recorded on the tabulated sheet
provided.
3. The steel rule was also used to measure the dimensions and the mass was theoretically
calculated.
Results
Mass
Balance
System
Specimen No.
Spring
Balance
Trials
M (grams)
M1
M2
(g)
(g)
93.4
95.5
M3
(g)
95.1
Mave= M1+
M2+ M3
(g)
Absolute
Error (ΔM) =
Mmax - Mmin
(g)
2.1
1. Small Steel
94.67
Cylinder
2. Medium steel
247.8
245.7 247.6
247.03
2.1
cylinder
3. Large Steel
596.8
600.5 598.3
598.33
3.7
cylinder
4. Aluminium
89.7
92.3
91.2
91.07
2.6
cylinder
Table 1: The table shows the results obtained from weighing the specimens on the Spring balance.
Relative Error (%) =
ΔM (100)/ Mave
2.218
0.85
0.618
2.855
Instrument Specimen 1
Specimen 2
Specimen 3
Specimen 4
Height, h Diameter, d Height, h Diameter, d Height, h Diameter, d Height, h Diameter, d
Vernier
25 mm
24.95mm
45.25 mm
29.9 mm
60.1 mm
40.25mm
46.8 mm
30.2mm
Caliper
Steel Rule
25 mm
25 mm
45 mm
30 mm
60 mm
40 mm
45 mm
30 mm
Table 2: The table shows the results obtained from measuring the diameter and height of the specimens
using the steel rule and the Vernier caliper.
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Specimen No.
Volume
Calculation
from Vernier
Caliper, VVC
(m3)
=π*(d/2)2
Volume
Calculation
from Steel
Rule,
VSR
(m3)
= π*(d/2)2
Mass
calculation
from Vernier
Caliper,
MVC
(kg)
=ρ mild steel VVC
= ρ Al VVC
Mass
calculation
from steel
rule,
MSR
(kg)
= ρ mild steel VSR
= ρ Al VSR
Mave =
MVC +
MSR
(kg)
Absolute
Error (ΔM)
=
Mmax - Mmin
0.09592
0.09631
0.09612
1.222 × 10−5
1.227 × 10−5
1. Small solid
cylinder
3.177 x 10-5
3.181 x 10-5
0.24939
0.24971
0.24955
2. Medium
solid
cylinder
7.647 x 10-5
7.540 x 10-5
0.60029
0.59189
0.59609
3. Large solid
cylinder
3.352 x 10-5
3.181 x 10-5
0.09084
0.08621
0.08853
4. Aluminium
cylinder
Table 3: Shows the calculated mass from obtaining the parameters of the specimens.
1.963 x 10-4
0.204
1.6 x 10-4
0.064
2.1 x 10-3
0.352
2.315 x 10-3
2.61
Sample Calculation:
The calculations shown below are just for the first column only. Others are calculated in the same
manner.
Volume Calculation for Vernier Caliper, VVC (m3)
 =  2 ℎ = (0.012475)2 (0.025) = 1.222 × 10−5 3
Volume Calculation for Steel Rule, VSR (m3)
 =  2 ℎ = (0.0125)2 (0.025) = 1.227 × 10−5 3
Mass calculation for Vernier Caliper, MVC (kg)
 =    = 7850 × (1.222 × 10−5 ) = 0.095927 
Mass calculation for steel rule, MSR (kg)
 =    = 7850 × (1.227 × 10−5 ) = 0.0963195 
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Relative
Error (%)
= ΔM
(100)/
Mave
Average mass, mav (kg)
 =
0.095927 + 0.0963195
= 0.09612325 
2
Absolute Error (ΔM)
∆ =
 −  0.0963195 − 0.095927
=
= 1.9625 × 10−4 
2
2
Relative Error (%)
% =
∆
1.9625 × 10−4
× 100 =
× 100 = 0.204%

0.09612325
Discussion of Results
Absolute error is the difference in measurement between an actual value and the experimental value, or
value that we get in any measurement. Absolute error is useful in every measurement if we want to get
the accurate value, and is practically less than one. For example, if the true weight of a packet of rice is 1
kg and we reweigh it and get 0.98 kg, then our absolute error would be ((1.00kg – 0.98)/2) = ±0.01 kg). In
our experiment, the absolute error calculated for mass of one of the specimen is ±.  × − 
or ±0.00019625 kg, which mean the accurate value falls in this range.
There are precautions to be taken to reduce absolute errors when doing experimental measurements. In
our experiment, we measured the heights, diameters and the masses of the four specimens with different
methods of measurements. For masses, we used a spring balance, a Vernier caliper and a 30 cm steel rule.
For height and diameters, we only used the Vernier caliper and the 30 cm steel ruler. Using the spring
balance, we took three (3) readings for mass of each specimen and the average was calculated, and using
the Vernier caliper we took two (2) readings for height, diameter for each specimen, form which the
masses were also calculated. Therefore, the precautions we took to reduce the absolute error were;
 We made sure that the instruments were correctly calibrated
 Three readings were recorded for mass and the average was taken (each specimen), and
 Two readings were recorded for height and diameter of each specimen using two
different measuring instruments (a steel ruler & a Vernier caliper)
The accuracy limit between the weighing instrument ranged from between 2 and 4 grams. The large errors
in the weighing accuracy was due to not calibrating the equipment precisely before measuring the
specimens.
During the experiment, only one mass balance system was employed. The Spring balance was used to
measure the mass of the four specimens. In comparison, the Vernier caliper and a steel rule was used to
measure parameters of the specimen and the mass was theoretically calculated. Therefore, a
comparison was drawn between the spring balance and the Vernier caliper and steel rule. After
calculation, by comparing the result from the two tables, the relative errors from the mass calculated
6|Page
using the Vernier caliper were less than those of the mass balance system. So for measuring mass, using
Vernier caliper and steel rule provide better accuracy than the mass balance system used.
Measuring the parameters before calculating the volume then mass gives better accuracy because the
limits of accuracy of the measuring instruments (Vernier caliper and steel rule) are very small which give
better precision that result in less error. Despite the precisions, the results varied slightly due to the fact
that the calculated mass of the specimen used densities of common material and not the densities of
the specific material in which the specimens were made of.
From the reviewing the result from only the spring balance, a relationship is evident in the result as the
limit accuracy varies with the mass. The lightest specimen (Aluminum) seems to possess the largest
relative error with 2.855%. It is then followed by the second lightest which is small steel cylinder with
2.218. The most accurate value in the comparison is the heaviest (largest steel cylinder = 0.618%)
Summary
Every physical thing has mass which in simple definition is the amount of matter present in a
body. Three instruments were used to measure mass which is the spring balance, Vernier
caliper and the steel ruler. The spring balanced was used to measure the experimental mass.
The theoretical mass was calculated using the product of density and volume after measuring
the parameters of the specimen.
Using both method of measuring mass as stated above, the results showed closed
approximation. We were able to determine the mass of the non-standard specimen.
The close approximation was represented by using accuracy limits and errors. The spring
balance had seemingly high errors compared to the theoretically calculated errors meaning a
precise measurement could not be obtained. Therefore, in every calculation the approximated
value of the measurement is paired with the errors to give at least a limits of precision in the
final measurement.
Overall the experiment was successfully, we were able to obtain all our results, however there
were some errors made during the experiments which were mentioned in our discussion. Every
experiment carried out, there were always errors made, therefore we took extra precautions
when carrying out experiment in order to obtain best results and to avoid errors. Apart from
the errors and the results discussed, the procedure were explained in simple English which is
self-explained and can be understood. All the results were tabulated on the table above, and
also the formula was provided. All the other necessary calculation was also done and tabulated
on the table above. All in all, the lab was a success but the quality of the results was hindered
by the state of the measuring instrument specifically the spring balance.
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Bibliography


https://en.wikipedia.org/wiki/Spring_scale
https://www.google.com/search?client=opera&q=density+of+mild+steel&sourceid=opera&ie=U
TF-8&oe=UTF-8
Appendices
Densities used:


Mild Steel = 7850 kg/m3
Aluminium Alloy 1100-H14 (99% Al) = 2710 kg/m3
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