ME 315 LABORATORY MANUAL
Compiled by
Engr. Dr. Patrick U. Akpan
Engr. Dr. Chigbogu G. Ozoegwu
Engr. Ifeanyi Jacobs
PREFACE
This material was borne out of a need, for undergradute students of the department of Mechanical
Engineering, University of Nigeria to have a good Laboratory manual that could serve as a guide
for the study of a course entitled ME 315 –Mechanical Engineering Laboratory I. It was prepared
specifically with these students in mind even though It can still be used in other schools.
The organization of this manual is as follows: Chapter 1 introduces the student to the course ME
315 –Mechanical Engineering Laboratory I. In the remaining chapters, various experiments are
looked at. These experiment are ; the wall Jib crane, simple screw jack, compound pendulum,
reactions of a beam and belt, Gyroscope, Weston Differential Pulley, Flywheel, Centrifugal force,
Torsion of rod, impact testing, Hardness testing, tensile testing . The objectives, procedures and
theory behind each experiment is clearly stated in this work.
The authors are grateful to Engrs. Ibe, Ajibo, Arthur Chinedu, Uduma, Martins for their assistance
in the preparation of this manual.
This laboratory course manual is geared towards helping undergraduate students of the department
to bridge the gap that exist between theory and practice.
The enthusiastic student is strongly advised to consult the cited reference materials for more
detailed information on the subject concerned as this manual is just a guide for the student. We
hope that it would enhance and not obfuscate the reader.
Engr. Dr. Patrick Udeme-obong Akpan
Engr. Dr. Chigbogu Ozoegwu
Engr. Ifeanyi Jacobs
August, 2021.
Contact: Patrick.akpan@unn.edu.ng; +2348102475639
ii
TABLE OF CONTENTS
PREFACE ................................................................................... ii
TABLE OF CONTENTS ................................................................... iii
LIST OF TABLES ......................................................................... vii
LIST OF FIGURES ....................................................................... viii
1 . INTRODUCTION ....................................................................... 1
1.1 Introduction & List of Experiments .......................................................... 1
1.2 Reference Materials ............................................................................ 2
1.3 Laboratory Schedule ........................................................................... 3
1.4 Laboratory Policies ............................................................................. 4
1.5 Assessment Method ............................................................................. 4
1.6 A Suggested Format for the Presentation of Laboratory Reports ....................... 4
1.7 Error Analysis.................................................................................... 5
2 WALL JIB CRANE ....................................................................... 7
2.1 Introduction ..................................................................................... 7
2.2 Apparatus Description ......................................................................... 7
2.3 Theory ............................................................................................ 8
2.3.1 Vector Resolution ......................................................................... 9
2.3.2 Polygon law of forces ................................................................... 10
2.4 Wall Jib Crane 1 Experiment................................................................ 11
2.4.1 Test Procedure........................................................................... 11
2.4.2 Results and Discussions ................................................................. 11
2.5 Wall Jib Crane 2 Experiment................................................................ 13
2.5.1 Test Procedure........................................................................... 13
2.5.2 Results and Discussions ................................................................. 13
3 3 SIMPLE SCREW JACK ............................................................... 15
3.1 Introduction ................................................................................... 15
3.2 Apparatus Description ....................................................................... 15
3.3 Theory .......................................................................................... 16
3.4 Test Procedures ............................................................................... 18
3.5 Results and Discussions ...................................................................... 19
4 THE SIMPLE AND COMPOUND PENDULUM ......................................... 20
4.1 Introduction ................................................................................... 20
4.2 Apparatus Description ....................................................................... 21
4.3 Theory .......................................................................................... 21
4.4 Simple Pendulum Experiment .............................................................. 27
4.4.1 Test Procedure........................................................................... 27
4.4.2 Results and Discussions ................................................................. 27
4.5 Compound Pendulum Experiment .......................................................... 28
4.5.1 Test Procedure........................................................................... 28
4.5.2 Results and Discussions ................................................................. 29
5 5 REACTION OF A BEAM ............................................................. 30
5.1 Introduction ................................................................................... 30
5.2 Apparatus description........................................................................ 30
5.3 Theory .......................................................................................... 31
5.4 Reaction of beam 1........................................................................... 32
iii
5.4.1 Test procedure...........................................................................
5.4.2 Results and Discussions .................................................................
5.5 Reaction of Beam 2 ..........................................................................
5.5.1 Test Procedure...........................................................................
5.5.2 Results and Discussion ..................................................................
6 BELT FRICTION ....................................................................... 36
6.1 Introduction ...................................................................................
6.2 Apparatus description........................................................................
6.3 Theory: .........................................................................................
6.4 Belt Friction 1: Flat Belt.....................................................................
6.4.1 Test Procedure...........................................................................
6.4.2 Results and Discussions .................................................................
6.5 Belt Friction 2: Vee-Belt.....................................................................
6.5.1 Test Procedure...........................................................................
6.5.2 Results and Discussions .................................................................
7 FLYWHEEL ............................................................................ 42
7.1 Introduction ...................................................................................
7.2 Apparatus description........................................................................
7.3 Theory ..........................................................................................
7.4 Test Procedure (extracted from [1]) ......................................................
7.5 Results .........................................................................................
7.6 Discussions .....................................................................................
8 WESTON DIFFERENTIAL PULLEY .................................................... 47
8.1 Introduction ...................................................................................
8.2 Apparatus description........................................................................
8.3 Theory ..........................................................................................
8.4 Test Procedures ...............................................................................
8.5 Results and Discussions ......................................................................
8.6 Conclusions ....................................................................................
9 GYROSCOPE ........................................................................... 51
9.1 Introduction ...................................................................................
9.2 Apparatus ......................................................................................
9.3 Theory ..........................................................................................
9.4 Determination of Moment of Inertia of the Armature and Disc ........................
9.5 Gyroscope 1 (Constant Torque (𝑻) Test) .................................................
9.5.1 Test Procedure...........................................................................
9.5.2 The results and Discussion .............................................................
9.6 Gyroscope 2 (Constant Precession rate (𝝎𝒑) Test) ......................................
9.6.1 Test Procedure...........................................................................
9.6.2 Results and Discussion ..................................................................
9.7 Gyroscope 3 (The constant rotor speed (𝝎) Test) .......................................
9.7.1 Test Procedure...........................................................................
9.7.2 Results and Discussion ..................................................................
10 CENTRIFUGAL FORCE .............................................................. 62
10.1 Introduction and Theory ...................................................................
10.2 Apparatus ....................................................................................
10.3 Centrifugal force 1 (varying Ma, constant Mb & r) ......................................
10.3.1 Test Procedure .........................................................................
32
33
34
34
34
36
36
37
37
37
38
39
39
40
42
42
43
45
46
46
47
47
48
49
50
50
51
51
53
54
57
57
57
58
58
59
60
60
60
62
63
64
64
iv
10.3.2 Results and Discussions................................................................
10.4 Centrifugal force 2 (varying Mb, constant Ma & r) ....................................
10.4.1 Test Procedure .........................................................................
10.4.2 Results and Discussions................................................................
10.5 Centrifugal force 3 (varying r, Constant Mb & Ma) .....................................
10.5.1 Test Procedure .........................................................................
10.5.2 Results and Discussions................................................................
11 TORSION OF ROD ................................................................... 68
11.1 Introduction ..................................................................................
11.2 Apparatus ....................................................................................
11.3 Theory ........................................................................................
11.4 Torsion 1 Experiment .......................................................................
11.4.1 Test procedure .........................................................................
11.4.2 Results and discussions ................................................................
11.5 Torsion 2......................................................................................
11.5.1 Test Procedure .........................................................................
11.5.2 Results and discussions ................................................................
12 IMPACT TESTING .................................................................... 74
12.1 Introduction ..................................................................................
12.2 Apparatus ....................................................................................
12.3 Theory ........................................................................................
12.4 Test Procedure ..............................................................................
12.5 Results and Discussions .....................................................................
12.6 Conclusions...................................................................................
13 HARDNESS TESTING ................................................................ 81
13.1 Introduction ..................................................................................
13.2 Apparatus ....................................................................................
13.3 Theory ........................................................................................
13.4 Test Procedure ..............................................................................
13.5 Results and Discussions .....................................................................
13.6 Conclusion ....................................................................................
14 TENSILE TESTING ................................................................... 87
14.1 Introduction ..................................................................................
14.2 Apparatus ....................................................................................
14.3 Theory ........................................................................................
14.4 Test Procedures .............................................................................
14.5 Results and Discussions .....................................................................
14.6 Conclusions...................................................................................
Appendix A Difference between Flat and Vee belt ..........................................
Appendix B Data for Tensile Testing ...........................................................
64
65
65
65
66
66
67
68
68
69
70
70
71
72
72
73
74
74
76
78
79
80
81
81
82
84
85
86
87
87
88
91
92
93
94
95
v
LIST OF TABLES
Table 1-1 List of experiments to be carried out ............................................................................ 1
Table 1-2 Reference Materials ...................................................................................................... 2
Table 1-3 Roster for the first Batch of Experiments...................................................................... 3
Table 1-4 Roster for the Second Batch of Experiments ....................................................... 3
Table1-5 Reporting format and breakdown of marks .................................................................. 5
Table 2-1 Sample data Sheet for Wall Jib Crane 1 Experiment ................................................... 12
Table 2-2 Sample data sheet for Wall Jib Crane 2 Experiment ................................................. 13
Table 3-1 Sample data sheet for simple screw Jack experiment ................................................ 19
Table 4-1 Sample data sheet for Simple Pendulum experiment ................................................ 27
Table 4-2 Sample data sheet for compound pendulum experiment .......................................... 29
Table 5-1 Sample data sheet for Beam reaction 2 experiment .................................................. 34
Table 5-2 Sample data sheet for Beam reaction 2 experiment .................................................. 35
Table 6-2 Sample data sheet for peg Angle 30°. ......................................................................... 38
Table 6-3 Sample data sheet for determination of co-efficient of friction ................................ 38
Table 6-4 Sample data sheet for peg Angle 30°. ......................................................................... 40
Table 6-5 Sample data sheet for determination of co-efficient of friction ................................ 40
Table 7-1 Sample data sheet for the ........................................................................................... 46
Table 8-1 Sample data sheet for differential pulley block experiment....................................... 50
Table 9-1 Sample data sheet for determining the moment of inertia of the armature ............. 56
Table 9-2 Sample data sheet for constant torque test in clockwise mode ................................. 58
Table 9-3 Sample data sheet for constant Precession speed test in clockwise mode ................ 59
Table 9-4 Sample data sheet for constant rotor speed test in clockwise mode ......................... 61
Table 10-1 Sample data sheet for Centrifugal force 1 experiment ............................................ 64
Table 10-2 Data Sheet for Centrifugal force 2 Experiment ........................................................ 66
Table 10-3 Data Sheet for Centrifugal force Experiment 3 ......................................................... 67
Table 11-1 Sample data sheet for Torsion of Rod ....................................................................... 71
Table 11-2 Sample data sheet for Torsion of Rod ....................................................................... 73
Table 12-1 Scales on the Impact testing Apparatus .................................................................... 74
Table 12-2 Standard data sheet for reporting Impact test of a given material .......................... 79
Table 13-1 Various Rockwell Hardness Scales ............................................................................ 84
Table 13-2 Data Sheet for test Specimen A ................................................................................ 85
Table 14-1 Sample data sheet for tensile testing of material. ................................................... 92
vii
LIST OF FIGURES
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
2.1 Wall Jib Crane apparatus .......................................................... 8
2.2 The interacting forces on the system under static equilibrium ............. 9
2.3 The Polygon of the forces acting on Point Z ................................... 11
3.1 Simple Screw Jack Experimental Apparatus ................................... 16
4.1 Compound Pendulum .............................................................. 22
4.2 A sample of the graph of T vs. h. For a compound Pendulum ............... 26
5.1 Beam Reaction Apparatus ......................................................... 31
5.2 Beam subjected to loading ....................................................... 32
5.3 Free body diagram for Beam reaction 1 ........................................ 33
6.1 Belt Friction experiment apparatus ............................................. 36
7.1 The flywheel experiment apparatus set-up .................................... 43
7.2 Fly-wheel working principle: work-energy ..................................... 44
8.1 Weston Differential Pulley Apparatus .......................................... 47
8.2 A schematic of a Weston differential Pulley Block ............................ 48
9.1 The gyroscope apparatus.......................................................... 52
9.2 Speed control unit ................................................................. 53
9.3 The spinning disc ................................................................... 53
9.4 The rotating mass (armature motor & Disc) suspended by a wire .......... 55
10.1 Schematic diagram of the Centrifugal Apparatus ........................... 62
10.2 Centrifugal Apparatus ............................................................ 63
11.1 Torsion testing apparatus........................................................ 69
12.1 Schematic diagram of an impact testing machine ........................... 75
12.2 Difference between Charpy and Izod Impact test ........................... 75
12.3 Principle of Impact testing using height measurements .................... 77
12.4 Principle of Impact testing using angle of fall and rise measurements ... 77
12.5 A Sample of the bar chart for impact testing results ........................ 80
13.1Rockwell hardness testing machine ............................................. 82
13.2 Figure Force-depth diagram of Rockwell test ................................ 83
13.3 A Sample of the bar chart ....................................................... 86
14.1 The Universal t Test Apparatus ................................................. 88
14.2 Test Specimen for Tension Test ................................................ 88
14.3 Stress-strain diagram ............................................................. 89
14.4 Method for determining the yield strength of a material ................... 91
viii
1 . INTRODUCTION
1.1 Introduction & List of Experiments
Mechanical Engineering Laboratory (ME 315) is a third year laboratory course offered by
Mechanical Engineering students. The course is geared towards deepening the practical
understanding of the students on subject areas such as: Mechanical Testing of materials;
Solid mechanics (statics & dynamics), Mechanics of machine; and etc. Students are expected
to complete a set of twenty experiments listed Table 1-1.
Table 1.1 shows a list of twenty experiments and the instructors scheduled for this academic
session.
Table 1-1 List of experiments to be carried out
No
Experiment Title
Instructors
Venue
Batch 1 Experiments
1
2
3
4
5
6
7
8
9
10
Wall Jib Crane 1
The Simple Screw Jack
The Simple Pendulum
Reaction of a Beam 1
Belt Friction 1: Flat-belt
Centrifugal force 1:
Gyroscope 1
Torsion 1
Impact testing: Charpy
Rockwell Hardness testing
Lab.
Lab.
Lab.
Lab.
Lab.
Lab.
Lab.
Lab.
Lab.
Lab.
11
12
13
14
15
Wall Jib Crane 2
Compound pendulum
Reaction of Beam 2
Belt Friction 1: V-belt
Flywheel
Lab.
Lab.
Lab.
Lab.
Lab.
16
17
18
19
20
Weston Differential Pulley
Centrifugal Force 2 & 3
Gyroscope 2
Torsion 2
Tensile testing
Lab.
Lab.
Lab.
Lab.
Lab.
Batch 2 Experiments
1
1.2 Reference Materials
The materials that were used in preparing this manual are listed in Table 1.3. The student
is strongly advised to consult these materials and any other material for further reading.
Table 1-2 Reference Materials
Title
Author (s)
Mechanical Engineering Design (6th Edition). McGraw Hill Shigley J. E and Mischke, C.
Publishers
R.
Vector Mechanics for Engineers – Statics and D Beer, F. P. and Johnston, E.
ynamics (6th Edition).
R
Mcgraw Hill Publishers
Laboratory/Instructional
Manual
for
Mechanical Mr Obi G. U.
Engineering Laboratory I(ME 315). Unpublished.
Mechanical Design and CAD Laboratory Manual. 2006 Universiti Tenaga Nasional.
Unpublished
Practical physics: New Age Internation (P) Ltd, New
Delhi
R.K. Shukla, Anchal
Srivatsava.
2
1.3 Laboratory Schedule
The schedule for the laboratory sessions are shown on Table 1-3 and Table 1-4. The student
is expected to complete three experiments per week. The time for the experiments are 15 pm daily for each experiment.
Table 1-3 Roster for the first Batch of Experiments
Contact
Session
A
B
C
D
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
1
10
9
8
7
6
5
4
3
2
2
1
10
9
8
7
6
5
4
3
3
2
1
10
9
8
7
6
5
4
4
3
2
1
10
9
8
7
6
5
Groups
E
F
G
Experiment Number
5
4
3
2
1
10
9
8
7
6
6
5
4
3
2
1
10
9
8
7
7
6
5
4
3
2
1
10
9
8
H
I
J
8
7
6
5
4
3
2
1
10
9
9
8
7
6
5
4
3
2
1
10
10
9
8
7
6
5
4
3
2
1
Table 1-4 Roster for the Second Batch of Experiments
Contact
Session
A
B
C
D
11th
12th
13th
14th
15th
16th
17th
18th
19th
20th
11
20
19
18
17
16
15
14
13
12
12
11
20
19
18
17
16
15
14
13
13
12
11
20
19
18
17
16
15
14
14
13
12
11
20
19
18
17
16
15
Groups
E
F
G
Experiment Number
15
14
13
12
11
20
19
18
17
16
16
15
14
13
12
11
20
19
18
17
17
16
15
14
13
12
11
20
19
18
H
I
J
18
17
16
15
14
13
12
11
20
19
19
18
17
16
15
14
13
12
11
20
20
19
18
17
16
15
14
13
12
11
3
1.4 Laboratory Policies
(A) Attendance:
(i)
(ii)
Attendance will be taken every time the class meets.
Any student arriving 20 minutes after the Laboratory session has started will not
be allowed to participate in the lab. & Examination for that day.
(iii)
Students will not be permitted to leave the during laboratory and examination
except for extreme emergencies.
(iv)
Regular students will be expected to complete at least 75% attendance in
Laboratory sessions to qualify for a final grade.
(v)
Carry-over/ external students will be expected to complete at least 40%
attendance in Laboratory to qualify for a final grade.
(vi)
You cannot report any experiment you did not perform.
1.5 Assessment Method
The course assessment would be solely based on the student’s participation during the
laboratory sessions and the examinations/reports that are submitted.
1.6 A Suggested Format for the Presentation of Laboratory Reports
Technical reports are usually written in the third person and laboratory report will take the
same form. All expressions should be in complete sentences and repetitions should be
avoided. A report should be concise. Wherever necessary, diagrams should be used to
communicate ideas. Each report should be written on paper of a uniform size and neatly
held together. Table 1.2 shows a list of items to include in the report and the percentage
contribution of each segment to the overall score.
4
Table1-5 Reporting format and breakdown of marks
S/No
1
2
3
4
5
6
Item
Description
Marks
obtainable
(%)
Title
The title of the experiment is stated
1%
Objectives
The objectives of the experiment should be clearly 2 %
stated.
Theory
& • The problems are analysed.
5 %
Analysis
• The relevant assumptions which have been made
are carefully stated.
• Relationships which are needed for later
calculations are derived.
Procedure
12%
• The method of carrying out the test is
described.
• A well annotated diagram of the apparatus is
sketched and is briefly described
• Safety precautions taken
Results and • Tabular presentation of data and the 75 %
Discussion
calculations.
• Calculations are done in this section.
• Graphical display of results whenever suitable.
• Discussion/Interpretation of results
Conclusion
A summary of the conclusions is made.
TOTAL MARKS (%)
5%
100%
Do not exceed 8 front pages only for each report (excluding graph sheets) while reporting
each experiment. Failure to adhere to this instruction will attract a penalty of 3% per extra
page written in each experiment.
1.7 Error Analysis
The deviations of measured data from the theoretically predicted value can be expressed
using different kinds of indicators namely: (a) The percentage error relative to the
5
theoretical value; (b) The percentage absolute error relative to the theoretical value; (c)
Mean percentage error (𝑀𝑃𝐸); and (d) Absolute mean percentage error (|𝑀𝑃𝐸|).
The percentage error relative to the measured value is given as:
% 𝐸𝑟𝑟𝑜𝑟𝑖 = 100
𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑𝑖 − 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑𝑖
𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑𝑖
(0.1)
The percentage absolute error relative to the measured value is given as
% |𝐸𝑟𝑟𝑜𝑟𝑖 | = 100
|𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑𝑖 − 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑𝑖 |
|𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑𝑖 |
(0.2)
The average of % Error𝑖 also called mean percentage error (𝑀𝑃𝐸) and the average of
% |Error𝑖 | also can be called absolute mean percentage error (|𝑀𝑃𝐸|), respectively
becomes.
𝑛
1
̅̅̅̅̅̅̅̅ = ∑ % 𝐸𝑟𝑟𝑜𝑟𝑖
% 𝐸𝑟𝑟𝑜𝑟
𝑛
(0.3)
𝑖=1
𝑛
̅̅̅̅̅̅̅̅̅̅ = 1 ∑ % |𝐸𝑟𝑟𝑜𝑟𝑖 |
%|𝐸𝑟𝑟𝑜𝑟|
𝑛
(0.4)
𝑖=1
6
2 WALL JIB CRANE
2.1 Introduction
A Wall Jib Crane is a load lifting device mounted on a wall. The self-evident confluence of
the four forces at the end of a wall jib Crane illustrates the application of a triangle of
forces so clearly to a real situation that this is an invaluable lesson.
The experiment enables the nature and relative magnitudes of the forces in the members
of the jib-crane to be found. The tension in the tie and thrust in the jib of the crane under
a known load are directly read from the spring scales attached. By taking the configurations
of the crane for a given load (measure the lengths of the members and then draw to scale).
These forces are also found graphically from the polygon of forces. The two sets of values
are tabulated and compared.
The objectives of this apparatus.
a. Comparison of measured forces with a polygon of forces and theoretical values
b. Comprehension of the action of the crane cable forces on the jib and the effect of
the jib inclination.
2.2 Apparatus Description
The wall mounted crane consists of a compression jib, and a tension tie which is adjustable,
and both incorporate a linear, direct reading spring balance to measure the forces in them.
A load hanger applies loads through the junction of the two. It folds flat when not in use.
Jib and ties re-adjustable to the original length. A set of weights is made available.
7
Figure 2.1 Wall Jib Crane apparatus
Two experiments can be carried out using this apparatus. The procedure for Wall Jib Crane
experiments 1 and 2 are stated in sub-sections 2.4 and 2.5 respectively.
2.3 Theory
Figure 2.2 shows the interaction of forces on the wall jib crane. The two approaches that
can be used to analyse the Static equilibrium at a point are: (a) vector resolution; and (b)
The polygon of forces.
8
Z
(b) Forces acting at point Z
(a) The configuration of the WJC
Figure 2.2 The interacting forces on the system under static equilibrium
2.3.1 Vector Resolution
The vector resolution approach requires that the resultant of the forces in both the vertical
and horizontal directions are zero. This is symbolically expressed thus
∑ 𝐹𝑖,𝑥 = 0
(1.1)
𝑖
∑ 𝐹𝑖,𝑦 = 0
(1.2)
𝑖
The configuration of the Wall Jibe Crane is given in Figure 2.2 (a) from which it is seen
that
𝜃𝑇 = cos−1 {
𝜃𝐽 = cos−1 {
𝐿2𝑇 + 𝐿21 − 𝐿2𝑆
}
2𝐿 𝑇 𝐿1
𝐿22 + 𝐿2𝐽 − 𝐿2𝑆
}
2𝐿2 𝐿𝐽
𝜃𝑆 = 𝜃𝐽 + cos −1 {
𝐿2𝑆 + 𝐿2𝐽 − 𝐿22
}
2𝐿𝑆 𝐿𝐽
(1.3)
(1.4)
(1.5)
The arising force system is shown in Figure 2.2 (b). The system of forces is in static
equilibrium thus, the balance in the vertical and horizontal directions respectively are:
9
∑ 𝐹𝑖,𝑦 = 𝐹𝑇 cos 𝜃𝑇 + 𝐹𝐽 cos 𝜃𝐽 − 𝐹𝑆 cos 𝜃𝑆 − 𝐹𝑊 = 0
(1.6)
𝑖
∑ 𝐹𝑖,𝑥 = −𝐹𝑇 sin 𝜃𝑇 + 𝐹𝐽 sin 𝜃𝐽 − 𝐹𝑆 sin 𝜃𝑆 = 0
(1.7)
𝑖
Equation (2.6) and (2.7) is solved simultaneously by elimination as follows (Equation
(2.6)× sin 𝜃𝑇 ) +((Equation 2.7)× cos 𝜃𝑇 ) and solving for 𝐹𝐽 gives:
𝐹𝐽 =
sin(𝜃𝑇 + 𝜃𝑆 ) + sin 𝜃𝑇
sin(𝜃𝑇 + 𝜃𝐽 )
𝐹𝑊
(1.8)
Also solved simultaneously by elimination as follows (Equation (2.6)× sin 𝜃𝐽 ) - (Equation
(2.6)× cos 𝜃𝐽 ) and solving for 𝐹𝑇 gives:
𝐹𝑇 =
sin(𝜃𝐽 + 𝜃𝑆 ) + sin 𝜃𝐽
sin(𝜃𝑇 + 𝜃𝐽 )
𝐹𝑊
(1.9)
2.3.2 Polygon law of forces
The law of polygon of forces which states that “ if a number of forces acting simultaneously
on a particle be represented in magnitude and direction by the sides of a polygon taken in
order, then the resultant of all these forces may be represented in magnitude and direction,
by the closing side of the polygon, taken in opposite order” .
If any number of coplanar forces acting on a particle be represented in magnitude and
direction by the sides of a closed polygon, taken in order, they shall be in equilibrium.
Converse of polygon law of forces states that if any number of forces acting on a particle at
equilibrium, a closed polygon can be drawn whose sides represents these forces in both
magnitude and direction. The converse of polygon law of forces is true in the sense that if
any number of co-planar forces acting at a point are in equilibrium and if sides are drawn
parallel and proportional to the forces then the sides will form only one closed polygon. But
if the sides are drawn only parallel and not proportional to the forces then they cannot be
represented by the sides of such a polygon as any number of such polygons can be drawn.
10
In this experiment, the theoretical values of the tension and compression in the members
of the crane could be found using the law of the polygon of forces acting on point Z (see
Figure 2.3).
Figure 2.3 The Polygon of the forces acting on Point Z
The forces acting on that joint are from the Tie, jib, string and load. These forces are
represented by the symbols 𝐹𝑡 , 𝐹𝑗 , 𝐹𝑠 𝑎𝑛𝑑 𝐹𝑤 respectively.
2.4 Wall Jib Crane 1 Experiment
2.4.1 Test Procedure
The procedure for this experiment is stated below.
1. Ensure that the spring balance are properly fixed in between the fixed bar and
compression bar.
2. Apply a load (𝑊) on the free end and note the reading of tension and compression
from spring balances.
3. Take 8 or 10 readings by increasing weights on the hanger and take the reading of
the corresponding value of tension as well as compression.
4. Measure the length of the members and draw to scale.
2.4.2 Results and Discussions
The readings of the experiment and the results of calculations are tabulated as shown in
Table 2-1;
11
Table 2-1 Sample data Sheet for Wall Jib Crane 1 Experiment
S/No
Measured
𝐹𝑊
(𝑁)
𝐹𝑇
(𝑁)
𝐹𝐽
(𝑁)
𝐿1
(𝑐𝑚)
𝐿2
(𝑐𝑚)
𝐿𝑇
(𝑐𝑚)
𝐿𝐽
(𝑐𝑚)
𝐿𝑆
(𝑐𝑚)
Calculated
using
relevant
equations
𝜃𝑇
𝜃𝑆
𝜃𝐽
(°) (°) (°)
Predicted using
graphical
method
𝐹𝑇,𝑡ℎ𝑒𝑜
(𝑁)
𝐹𝐽,𝑡ℎ𝑒𝑜
(𝑁)
1
2
3
4
5
6
7
Complete the underlisted tasks:
1. Find the theoretical value of tension (𝐹𝑇,𝑡ℎ𝑒𝑜 ) and compression (𝐹𝐽,𝑡ℎ𝑒𝑜 ) by using the
formulae given in theory and compare that value with actual values.
2. Find out the % of error in tension and compression.
3. Plot and discuss a graph of jib forces against the load; the measured and predicted
values should be plotted on same axis.
4. Plot and discuss a graph of the tie forces against the load; the measured and
predicted values should be plotted on same axis.
5.
Calculate the MPE and |MPE| for the jib and the tie and use the values to say how
the predictions agree with measurements. State the reasons for disagreements
(sources of error)
6. Is there any discrepancy between the theoretical value and the experimental results?
If a discrepancy exists, why does it exist?
12
2.5 Wall Jib Crane 2 Experiment
2.5.1 Test Procedure
1. Ensure that the spring balance are properly fixed in between the fixed bar and
compression bar.
2. Take a measurement of the distance between point A and C.
3. Divide the distance length AC into five equal intervals.
1
4. Move the adjuster/pin of the string to the first position i.e 𝐿1 = 5 𝐴𝐶 .
5. Insert a total weight that is between 2400g and 3400g on the load hanger.
6. Measure the dimensions of the configuration (𝐿1 , 𝐿2 , 𝐿 𝑇 , 𝐿𝐽 , 𝐿𝑆 ) and the forces in the
tie (𝐹𝑇 ) and the Jib (𝐹𝐽 ).
2
3
7. Repeat steps (iii)-(v) for the other three positions 𝐿1 = 5 𝐴𝐶 ; 𝐿1 = 5 𝐴𝐶 ; and 𝐿1 =
4
𝐴𝐶
5
. Note: the same weight selected at step (iv) above should be used for all the
positions.
2.5.2 Results and Discussions
The readings of the experiment and the calculated angles is tabulated as shown Table 2-2.
Table 2-2 Sample data sheet for Wall Jib Crane 2 Experiment
S/No
Measured
𝐹𝑊
(𝑁)
Calculated
using
relevant
equations
Predicted using
graphical
method
𝐹𝑇
𝐹𝐽
𝐿1
𝐿2
𝐿𝑇
𝐿𝐽
𝐿𝑆
𝜃𝑇 𝜃𝐽 𝜃𝑆 𝐹𝑇,𝑡ℎ𝑒𝑜
(𝑁) (𝑁) (𝑐𝑚) (𝑐𝑚) (𝑐𝑚) (𝑐𝑚) (𝑐𝑚) (°) (°) (°)
(𝑁)
𝐹𝐽,𝑡ℎ𝑒𝑜
(𝑁)
1
2
3
13
4
Complete the underlisted tasks:
1.
Determine the angles, 𝜃𝑇 , 𝜃𝐽 and 𝜃𝑆 for each readings taken, using the relevant
equations.
2.
Use the law of polygon of forces (graphical method) to determine the theoretical
values of 𝐹𝐽 and 𝐹𝑇 for each of the readings.
3.
Determine the absolute mean percentage error of values for 𝐹𝐽 and 𝐹𝑇 .
4.
Discuss some of the sources of the errors in the experiment.
14
3 3 SIMPLE SCREW JACK
Equation Chapter (Next) Section 1
3.1 Introduction
The Screw Jack is a simple device for raising a heavy load with comparatively little effort
and when only a short lift is required. The motor car jack is a typical example. It consists
essentially of a nut and screw in which the nut is held stationary while the screw is turned
and lifts the weight. The same principle is used in the bench vice and screw press.
A long lever such as a crowbar will also lift a heavy load through a short distance with only
a small effort. With the screw jack the friction of the screw is sufficient to hold the load in
the raised position when the effort is removed so that it will not run back or "overhaul". This
is an important feature of the screw jack.
The objectives of this experiment are to:
1. Measure the effort required to raise various loads using a simple form of screw jack.
2. Determine how the mechanical advantage, velocity ratio and efficiency varies with
load.
3. Test whether the screw jack "overhauls".
3.2 Apparatus Description
The simple screw jack used in this experiment (see Figure 3.1) is based around a benchmounted base incorporating a turntable fitted with a metric square pitch screw jack
thread. The apparatus is stood on a firm bench and a cord is wound around the periphery of
the turntable. The free end of the cord is threaded over a pulley and then hangs vertically
to accept the load hanger supplied. A set of calibrated weights is supplied which are
suspended from the load hanger thus producing a known torque on the system. To adjust
the experimental parameters further the calibrated weights can also be applied to the top
surface of the turntable
15
Figure 3.1 Simple Screw Jack Experimental Apparatus
3.3 Theory
Screw jack is used to raise heavy loads. The apparatus works on a simple principle of screw
and nut. The axial distance between the corresponding threads is known as pitch. Let this
pitch be 𝑃 and 𝑅 the effective radius of the turn-table (i.e allowing for string radius) on
which the load (𝑊) is to be placed and lifted.
If the table turns through one revolution, then the load rises in one revolution which is
equivalent to the length of the screw jack = 𝐿.
Since this is a single start jack
(2.1)
𝐿 = 𝑃𝑖𝑡𝑐ℎ = 𝑃
Distance moved by the effort (𝐸) is equivalent to 2𝜋𝑅
Velocity ratio(𝑽𝑹): Velocity ratio is defined mathematically as
𝑉𝑅 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑚𝑜𝑣𝑒𝑑 𝑏𝑦 𝑒𝑓𝑓𝑜𝑟𝑡
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑚𝑜𝑣𝑒𝑑 𝑏𝑦 𝑙𝑜𝑎𝑑
=
𝑥𝑒
𝑥𝐿
(2.2)
16
For one revolution of the turn table, VR is given in Equation. 3.3
2𝜋𝑅
𝐿
(2.3)
2𝜋𝑅
𝜋𝐷𝑒
=
𝑃
𝑃
(2.4)
𝑉𝑅 =
𝑉𝑅 =
Where 𝐷𝑒 is the effective diameter of the turn table.
Mechanical advantage (MA): is the factor by which a mechanism multiplies the force put
into it. It is the ratio of the force exerted by a machine (the output) to the force exerted
on the machine, usually by an operator (the input). The theoretical mechanical advantage
of a system is the ratio of the force that performs the useful work to the force applied,
assuming there is no friction in the system. In practice, the actual mechanical advantage
will be less than the theoretical value by an amount determined by the amount of friction.
𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑎𝑑𝑣𝑎𝑛𝑡𝑎𝑔𝑒 (𝑀𝐴) =
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 (𝜂) % =
𝑊
𝐸
𝑀𝐴
∗ 100
𝑉𝑅
(2.5)
(2.6)
The Law of Machine: The law of machine stems from the principle of work. It is an
expression which gives the relationship of the load (𝑊) and effort (𝐸) and is expressed thus
as:
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 (𝜂) % =
𝑀𝐴
∗ 100
𝑉𝑅
(2.7)
For a machine that is 100% efficient, mechanical advantage equals velocity ratio.
𝐸. 𝑥𝑒 = (𝑊 + 𝑊𝑓 )𝑥𝐿
(2.8)
17
Where 𝑊𝑓 , 𝑥𝑒 and 𝑥𝐿 are the friction load, distance moved by effort and distance moved
by load respectively.
Equation 3.8 can be re-written as
𝐸=𝑎𝑊+ 𝑏
(2.9)
Where 𝑎 and 𝑏 are constants to be determined from a graph of Effort versus load.
3.4 Test Procedures
Ensure that the threads are properly lubricated with light oil. Grease must not be used
because it causes sticky motion.
Phase 1 Determination of Theoretical Velocity Ratio (VR)
i.
Measure the lead (distance between two successive thread) of the screw jack with
the aid of a Vernier Caliper. Note: For a single start thread screw jack, the lead of
the jack is equal to the pitch.
ii.
Measure the effective diameter of the turn table with the aid of an outside caliper.
The effective diameter of the turn table is the summation of the diameter of the
turn table and the thread.
iii.
With the values obtained in (i) and (ii) , the theoretical V. R can be determined using
Eq. 3.4
Phase 2 Determination of Experimental Velocity Ratio (VR)
Obtain the experimental velocity ratio by measuring the actual distances moved by the
effort and the load during an operation thus getting an experimental answer. To achieve
this carryout the following.
i.
Ensure that the four or five threads are up before you continue.
ii.
At no load (W) measure the distance between the effort hanger and the floor (Z).
iii.
At no load (W) measure the initial load table distance (𝐵1 ).
iv.
Allow the load hanger to make contact with the floor. At this point, measure the
new load table distance (𝐵2 ).
The experimental V.R can be evaluated using Eq. 3.10
18
𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉. 𝑅 =
(2.10)
𝑍
𝐵2 − 𝐵1
Note: The values obtained for the theoretical VR and the experimental VR should be very
close and their mean can be taken as the VR.
Phase 3 Determination of Experimental Friction Load (𝑊𝑓 )
1. Measure the diameter of the turn table and that of the sting.
2. Wrap the string round the circumference of the flanged table and pass it over the
pulley. The free end of the string should be tied to the hanger where the weights
are to be hanged.
3. Measure the pitch of the thread with the help of the Vernier Calliper.
4. Commencing with a small load (W) on the screw head and some small weight (Effort)
on the hanger so that the load W is just lifted. The effort E is equal to the weight on
the hanger.
5. Repeat for a series of increasing loads (W) and find the corresponding efforts applied
for the consecutive readings. This should be done at least seven (7) times. (the
maximum load to place on the simple screw should not exceed 170 Kg)
6. Calculate mechanical advantage, velocity ratio and efficiency in each case.
3.5 Results and Discussions
Tabulate your readings using the format on Table 3-1.
Table 3-1 Sample data sheet for simple screw Jack experiment
Load (W)
Effort (E)
Mechanical
Advantage
(MA)
Velocity
ratio (VR)
Friction Load
(𝑊𝑓 )
Efficiency (𝜂)
Complete the underlisted tasks:
1. Plot and discuss the following graphs: (a) Efficiency vs. Load. (b) Effort vs. Load.
(c) Efficiency vs. Friction load.
2. Derive the law of the machine.
3. State the safety precaution.
19
4 THE SIMPLE AND COMPOUND PENDULUM
Equation Chapter (Next) Section 1
4.1 Introduction
The gravitational attraction of a body towards the center of the earth results in the
same acceleration for all falling bodies at a location, irrespective of their mass,
shape or material, and this acceleration is called the acceleration due to gravity, g.
The value of g varies from place to place, being greatest at the poles and the least
at the equator. Because this value is large, bodies fall quickly to the surface of the
earth when dropped, and so it is very difficult to measure their acceleration directly
with considerable accuracy.
Therefore, the acceleration due to gravity is often determined by indirect methods
– for example, using a simple pendulum or a compound pendulum. If we determine
g using a simple pendulum, the result is not very accurate because an ideal simple
pendulum cannot be realized under laboratory conditions. Hence, you will use a
simple and a compound pendulum to determine the acceleration due to gravity in
the laboratory.
A rigid body of any shape which is free to oscillate without any friction on a vertical plane
is called compound pendulum. It swings harmonically back and forth about a vertical z-axis.
The harmonic motion is due to gravitational force, which is directed downward, acting on
the pendulum.
The objectives of the experiment are:
(i)
To determine the acceleration due to gravity (g) at Nsukka.
(ii)
To verify that there are two pivot points on either side of the centre of
gravity (C.G.) about which the time period is the same.
(iii)
To determine the radius of gyration of a bar pendulum
(iv)
To compare the g, value obtained using the simple pendulum and
compound pendulum.
20
4.2 Apparatus Description
The compound pendulum consists of a metallic bar. A series of circular holes are made along
the length of the bar. The bar is suspended from a horizontal knife-edge passing through
any of the holes Figure 4.1. The knife edge, in turn, is fixed on a platform provided with
the screws. By adjusting the rear screw the platform can be made horizontal. Other
instruments used are the stop watch, meter rule.
4.3 Theory
A simple pendulum consists of a small body called a “bob” (usually a sphere) attached to
the end of a string the length of which is great compared with the dimensions of the bob
and the mass of which is negligible in comparison with that of the bob. Under these
conditions the mass of the bob may be regarded as concentrated at its center of gravity,
and the length of the pendulum is the distance of this point from the axis of suspension.
When the dimensions of the suspended body are not negligible in comparison with the
distance from the axis of suspension to the center of gravity, the pendulum is called a
compound, or physical, pendulum. A rigid body mounted upon a horizontal axis so as to
vibrate under the force of gravity is a compound pendulum.
The compound pendulum used in this experiment is a uniform bar suspended at different
locations along its length.
21
P
G
Figure 4.1 Compound Pendulum
If the bar is displaced from its equilibrium position by an angle (θ). In the equilibrium
position (G) the center of gravity (C.G) of the body is vertically below P (the point
of suspension). The distance GP is ℎ and the mass of the body is m. The restoring
torque for an angular displacement 𝜃 is given as:
𝜏 = 𝐼𝛼 = −𝑚𝑔ℎ 𝑠𝑖𝑛𝜃
(3.1)
Where I is moment of inertia of the body through the axis P and α is angular
acceleration. α =
𝑑2 𝜃
𝑑𝑡 2
For small amplitudes (θ ≈ 0),
𝜏 = 𝐼𝛼 = −𝑚𝑔ℎ 𝑠𝑖𝑛𝜃
(3.2)
𝑑2 𝜃
= −𝑚𝑔ℎ 𝜃
𝑑𝑡 2
(3.3)
𝐼
Equation (4.3) represents a simple harmonic motion and hence the time period of
oscillation is given by
22
(3.4)
𝐼
𝑇 = 2𝜋√
𝑚𝑔ℎ
Now 𝐼 = 𝐼𝐶𝐺 + 𝑚ℎ2 , where 𝐼𝐶𝐺 = 𝑚𝐾 2 , is the moment of inertia of the body about an axis
parallel with axis of oscillation and passing through the center of gravity (CG). where K is
the radius of gyration about the axis passing through CG. Thus
𝐾2
+ℎ
𝑚𝐾 2 + 𝑚ℎ2
𝑇 = 2𝜋 √
= 2𝜋√ ℎ
𝑚𝑔ℎ
𝑔
(3.5)
The time period of a simple pendulum of length L, is given by:
𝐿
𝑇 = 2𝜋√
𝑔
(3.6)
Comparing Eq. 4.5 and 4.6 we observe:
𝐿=
𝐾2
+ℎ
ℎ
(3.7)
L is the length of “equivalent simple pendulum”. If all the mass of the body were
concentrated at an imaginary point (lets say O) such that the distance between that point
and the point of suspension equals
𝐾2
ℎ
+ ℎ , we would have a simple pendulum with the same
time period. That point O is called the ‘Centre of Oscillation’. Time period T will have
minimum value when ℎ = 𝐾; 𝐿 = 2ℎ.
Now from Eq. (4.7)
ℎ2 − ℎ𝐿 + 𝐾 2 = 0
(3.8)
I.e. a quadratic equation in ℎ. Equation 4.8 has two roots ℎ1 and ℎ2 such that:
ℎ1 + ℎ2 = 𝐿
(3.9)
ℎ1 ℎ2 = 𝐾 2
(3.10)
23
Thus both ℎ1 and ℎ2 are positive. This means that on one side of C.G there are two positions
of the center of suspension about which the time periods are the same.
Similarly, there will be a pair of positions of the center of suspension on the other side of
the C.G about which the time periods will be the same. Thus, there are four positions of
the centers of suspension, two on either side of the C.G, about which the time periods of
the pendulum would be the same. The distance between two such positions of the centers
of suspension, asymmetrically located on either side of C.G, is the length L of the simple
equivalent pendulum. Thus, if the body were supported on a parallel axis through the point
O, it would oscillate with the same time period T as when supported at P. Now it is evident
that on either side of G, there are infinite numbers of such pair of points satisfying Eqns.
(4.9 and 4.10). If the body is supported by an axis through C.G, the time period of oscillation
would be infinite. From any other axis in the body the time period is given by Eq. (4.5).
From Eq.(4.6) and (4.10), the value of g and K are given by:
𝐿
𝑇2
(3.11)
𝐾 = √ℎ1 ℎ2
(3.12)
𝑔 = 4𝜋 2
Determination of Acceleration due to gravity (g) , Radius of Gyration (𝑲):
Two methods are available for the determination acceleration due to gravity (𝑔) and radius
of gyration(𝐾) from the experimental data. And these are:
Method A
If a graph of Period (T) vs. distance of point of suspension from C.G (h) is plotted. It is
expected to have a shape like the one in Fig.4.2 with two curves which are symmetrical
about the C.G of the bar. To find the length L of a simple pendulum with the same period
as a compound pendulum, a horizontal line 𝐵′ 𝐴′ AB can be drawn which cuts the graph at
points 𝐵′ , 𝐴′ , A and B all of which reads the same time period 𝑇1 . For 𝐵′ as the centre of
suspension, A is the centre of oscillation (A is at distance of ℎ1 + ℎ2
= L from the center
of suspension 𝐵′ ). Similarly, for 𝐴′ as the centre of suspension, B is the centre of oscillation.
Draw another horizontal line EFGH (see Figure 4.2) to cut the period at 𝑇2
24
Acceleration due to gravity, g
For line 𝐵′ 𝐴′ AB
𝐿=
For line EFGH
𝐵′ 𝐴 + 𝐴′ 𝐵
= ⋯ 𝑐𝑚
2
𝐿=
𝐸𝐺 + 𝐹𝐻
= ⋯ 𝑐𝑚
2
𝑇2 = ⋯ 𝑠𝑒𝑐
𝑇1 = ⋯ 𝑠𝑒𝑐
𝑔 = ⋯ 𝑐𝑚/𝑠 2
Hence using the formular for 𝑔 as given in
Eq.4.11
𝑔 = ⋯ 𝑐𝑚/𝑠 2
From the two 𝑔 values calculated, the mean value of 𝑔 = ⋯ 𝑐𝑚/𝑠 2 .
Radius of gyration, K
From Figure 4.2 for line 𝐵′ 𝐴′ AB, The values for ℎ1 and ℎ2 are shown and the radius of
gyration (𝐾1 ) can be calculated using Eq. 4.12. Use the same approach for line EFGH and
calculate the second radius of gyration (𝐾2 ). Hence the mean value for the radius of
gyration about the C.G is gotten by the calculating the arithmetic average of (𝐾1 ) 𝑎𝑛𝑑 (𝐾2 ).
25
E
G
F
H
C.G
Figure 4.2 A sample of the graph of T vs. h. For a compound Pendulum
Method B: Ferguson’s method
Alternatively the measurements can also be used to determine g and K using Ferguson’s
method as explained below. Manipulating Equations (4.5) we get:
ℎ2 =
𝑔
ℎ𝑇 2 − 𝐾 2 = 0
4𝜋 2
(3.13)
From Equation 4.13 a graph between ℎ2 and ℎ𝑇 2 should therefore be a straight line with
slope 𝑔⁄4𝜋 2 , The intercept on the y-axis is − 𝐾 2 .
Acceleration due to gravity; g = 4𝜋 2 × slope
Radius of gyration;
𝐾 = √(𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡)
26
4.4 Simple Pendulum Experiment
4.4.1 Test Procedure
1. Mount the simple pendulum, measure the required length of the string and with an
Allen key tighten the bob.
2. Tilt the simple pendulum slightly by an angle less than 5 degrees and allow it to
oscillate
3. Measure the time it takes for 30 oscillations.
4. For the same string length, repeat steps 2 and 3 and take the mean value.
5. Repeat steps 2 to 4 six times for different decreasing length of the string
4.4.2 Results and Discussions
The format for reporting the observations of the simple pendulum experiments are given
in Table 4-1.
Table 4-1 Sample data sheet for Simple Pendulum experiment
S/No
Length (𝑚)
(𝐿)
𝑇1
Time (𝑠)
𝑇1
𝑇𝑎𝑣
Period
(𝑇)
( 𝑇 2)
Complete the following tasks:
1. Plot and discuss a graph of 𝑇 2 𝑣𝑠. 𝐿.
2. Determine the acceleration due to gravity from the graph in (1) above.
3. State the reason(s) why you think that the acceleration due to gravity determined
using this method may be slightly inaccurate.
4. State the underlying assumptions used in the experiment
5. What is the acceleration due to gravity in a region where a simple pendulum having
a length 75 cm has a period of 1.7357 s?
6. A pendulum with a period of 2.00000 s in one location (g= 9.8m/s2) is moved to a
new location where the period is now 1.99796 s. What is the acceleration due to
gravity at its new location?
7. For a given simple pendulum
a. What is the effect on the period of a pendulum if you double its length?
27
b.
What is the effect on the period of a pendulum if you decrease its length by
5.00%?
8. An engineer builds two simple pendula. Both are suspended from small wires secured
to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1
has a bob with a mass of 10kg. Pendulum 2 has a bob with a mass of 100 kg. Will the
period of motion of both pendula be the same if the bobs are both displaced slightly
by an angle of 10°? State your reasons.
4.5 Compound Pendulum Experiment
4.5.1 Test Procedure
1. Measure the total length of the compound pendulum bar without the sliding weight
2. Place the sliding weight of the compound pendulum close to the last hole of the
pendulum bar.
3. Balance the bar on a sharp wedge and mark the position of its C.G.
4. Fix the knife edges in the outermost holes at either end of the bar pendulum. The knife
edges should be horizontal and lie symmetrically with respect to center of gravity of the
bar.
5. Suspend the pendulum vertically.
6. Tilt the bar slightly to one side of the equilibrium position and let it oscillate with the
amplitude not exceeding 5 degrees. Make sure that there is no air current in the vicinity
of the pendulum.
7. Use the stop watch to measure the time for 30 oscillations. The time should be measured
after the pendulum has had a few oscillations and the oscillations have become regular.
8. Measure the distance h from C.G. to the knife edge.
9. Record the results in Table 4-2. Repeat the measurement of the time for 30 oscillations
and take the mean.
10. Suspend the pendulum on the knife edge at other holes and repeat the measurements
in steps 6 -9 above.
28
4.5.2 Results and Discussions
The format for reporting the observations of the compound pendulum experiments are given
in Table 4-2.
Table 4-2 Sample data sheet for compound pendulum experiment
S/No
ℎ (𝑐𝑚)
(𝐿)
ℎ2
( 𝑐𝑚2 )
𝑇1
Time (𝑠)
𝑇1
𝑇𝑎𝑣
Period (𝑠)
(𝑇)
( ℎ. 𝑇 2 )
Complete the following task:
1. Plot and discuss the following graphs
•
𝑇 𝑣𝑠. ℎ (To be able to find g, K using method A)
•
ℎ2 vs. ℎ𝑇 2 ( To be able to find g, K using method B)
2. Discuss the observations in both graphs.
3. Calculate the moment of inertia of the compound pendulum about the centre of
gravity using results :
•
From method A (using 𝑇 = 1.3 𝑠𝑒𝑐𝑠)
•
From method B.
29
5 5 REACTION OF A BEAM
Equation Chapter (Next) Section 1
5.1 Introduction
The reactions at the support of a simple supported beam are directly measured by spring
balances in this experiment. These reactions are also calculated by noting the dimensions
of the beam. The loads suspended from it are at known positions and taking moments of the
forces about the supports in turn.
1. To apply a stable system of loading to a pivoted beam
2. To compare with values obtained from calculation using simple moments
5.2 Apparatus description
This equipment (Figure 5.1) provides a simple easy to understand experiment on the
equilibrium of moments. Several loads can be put on the beam at various positions. These
will make the beam rotate. The student has to determine the moment necessary to
overcome this rotation and keep the beam level. On a practical level, this principle is used
in the measurement of goods, such as in chemical balances and steel yards.
The beam is tied in each direction from the central pivot in cm. Three wire stirrups, weight
hangers and a set of weights are included.
30
𝑅𝑜
𝑅𝐸
Beam
𝐹1
𝐹2
𝐹3
Base Support
Figure 5.1 Beam Reaction Apparatus
5.3 Theory
The law of static equilibrium in a plane, of a system of forces, requires that every force is
equal in magnitude to the resultant of the rest but opposite in direction and every moment
is equal in magnitude to the resultant of the rest but opposite in sense. This is symbolically
expressed thus:
∑ 𝐹𝑖,𝑥 = 0
(4.1)
𝑖
∑ 𝐹𝑖,𝑦 = 0
(4.2)
𝑖
∑ 𝑀𝑖,𝑥 = 0
(4.3)
𝑖
∑ 𝑀𝑖,𝑦 = 0
(4.4)
𝑖
In our experimental equipment, 𝐹𝑖,𝑥 = 0 and 𝑀𝑖,𝑥 = 0. The moments are taken about the
centroidal axis chosen section of the beam
31
The arising force system is shown in Figure 5.2.
Figure 5.2 Beam subjected to loading
The system is in static equilibrium thus the balance forces and moments gives:
𝑅𝑂 = 𝐹1 (1 −
𝐿1
𝐿2
𝐿3
𝐿𝐶.𝐺.
) + 𝐹2 (1 − ) + 𝐹3 (1 − ) + 𝐹𝐵 (1 −
)
𝐿𝐸
𝐿𝐸
𝐿𝐸
𝐿𝐸
𝑅𝐸 =
1
(𝐹 𝐿 + 𝐹2 𝐿2 + 𝐹3 𝐿3 + 𝐹𝐵 𝐿𝐶.𝐺. )
𝐿𝐸 1 1
(4.5)
(4.6)
𝐿𝐶.𝐺 is the distance between the centre of gravity of beam and 𝑅𝑂 .
5.4 Reaction of beam 1
5.4.1 Test procedure
In this experiment, a single load hanger (including the load placed on it) is fixed (see
Figure 5.3) and the moment arm changes in steps of 10cm such that the locations of the
hanger are marked as 𝐻𝑖 .
The test procedure is described
1. Set up the apparatus as shown in Figure 5.3.
2. Record the distance of the load hanger and the spring balance from the left spring
balance measuring the reaction (𝑅𝑂 ).
3. The initial readings of the spring balances equals the weight of the unloaded load
hanger and the beam itself. This is taken as the zero readings.
32
4. Place a fixed load that is greater than 3.6kg on the load hanger at 10cm away from
the reaction 𝑅𝑂 , and take the readings both spring balances.
In increasing order, change the position of the load hanger for the remaining positions in
5. Table 5-1.
𝑂𝐻
Figure 5.3 Free body diagram for Beam reaction 1
Note: The theoretical values of the reactions can be determined by
𝑅𝐸,𝑖 =
1
(𝐹 ∙ 𝑂𝐻 + 𝐹𝐵 𝐿𝐶.𝐺. )
𝐿𝐸
𝑅𝑂,𝑖 = 𝐹 (1 −
𝑂𝐻
𝐿𝐶.𝐺.
) + 𝐹𝐵 (1 −
)
𝐿𝐸
𝐿𝐸
(4.7)
(4.8)
5.4.2 Results and Discussions
Tabulate the results in the format of Table 5.1 together with the loading scheme.
(i)
Plot and discuss a graph of the 𝑅𝑂.𝑒𝑥𝑝 and 𝑅𝑂.𝑐𝑎𝑙 versus 𝑂𝐻
(ii)
Plot and discuss a graph of the 𝑅𝐸.𝑒𝑥𝑝 and 𝑅𝐸.𝑐𝑎𝑙 versus 𝑂𝐻
(iii)
Calculate the mean percentage error (MPE) and the absolute mean percentage
error (|MPE|) for the 𝑅𝑜 and 𝑅𝐸 respectively.
(iv)
(v)
Use the calculated values to judge the accuracy of your analysis.
Discuss the sources of your errors.
Take g = 10m/s2
33
Table 5-1 Sample data sheet for Beam reaction 2 experiment
Constant load selected 𝐹 = ____________ 𝑁
𝑖
𝑂𝐻
𝑅𝑂.𝑒𝑥𝑝
𝑅𝐸.𝑒𝑥𝑝
𝑅𝑂.𝑐𝑎𝑙
𝑅𝐸.𝑐𝑎𝑙
(𝑐𝑚)
[𝑁]
[𝑁]
[𝑁]
[𝑁]
1
10
2
20
3
30
4
40
5
50
6
60
7
70
8
80
9
90
Note: 𝑅𝑂.𝑒𝑥𝑝 and 𝑅𝐸.𝑒𝑥𝑝 are the experimental data obtained 𝑅𝑂.𝑒𝑥𝑝 and 𝑅𝐸.𝑒𝑥𝑝 are the
calculated reactions.
5.5 Reaction of Beam 2
5.5.1 Test Procedure
6. Set up the apparatus as shown in Figure 5.2.
7. Record the distance of the load hangers and the spring balances from the left spring
balance measuring the reaction (𝑅𝑂 ).
8. The initial readings of the spring balances equals the weight of the unloaded load
hanger and the beam itself. This is taken as the zero readings.
9. In increasing order, adding different sizes of weights to the load hangers, so that the
new weight equals the summation of the weight and the weight of the load hanger.
Read the new values of the spring balance.
10. Repeat step 4 six times and record the values.
5.5.2 Results and Discussion
Tabulate the results in the format of Figure 5.2 together with the loading scheme.
Summarize your results in graphical plots of predicted versus measured forces. Calculate
the mean percentage error (MPE) and the absolute mean percentage error (|MPE|) for the
34
𝑅𝑜 and 𝑅𝐸 respectively. Use the calculated values to judge the accuracy of your analysis.
Discuss the sources of your errors.
Take g = 10m/s2
Table 5-2 Sample data sheet for Beam reaction 2 experiment
S/No
𝐹1
[𝑁]
𝐹2
[𝑁]
𝐹3
[𝑁]
𝑅𝑂.𝑒𝑥𝑝
[𝑁]
𝑅𝐸.𝑒𝑥𝑝
[𝑁]
𝑅𝑂.𝑐𝑎𝑙
[𝑁]
𝑅𝐸.𝑐𝑎𝑙
[𝑁]
35
6 BELT FRICTION
6.1 Introduction
Equation Chapter (Next) Section 1
The objectives of the experiment are to : (a) investigate the relationship between belt
tensions, angle of wrap and co-efficient of friction for flat and V belts; (b) determine the
effect of the angle of wrap to the power that can be transmitted for belt drive mechanism;
(c) determine the effect of the belt tensions to the power that can be transmitted for belt
drive mechanism; (d) compare the power transmission capability of flat and V-belt; (e)
compare the empirical data with the theoretically derived solutions
6.2 Apparatus description
The apparatus for the belt friction experiment (Figure 6.1) is made up of a pulley mounted
upon ball bearings, spring balance for measuring the tight side tension (𝑇2 ), belts (Vee and
Flat), a load hanger for carrying the slack slide weight (𝑇1 ), an angle marker for changing
the angle of wrap of the belt and a peg for fastening the angle marker to the wall.
Figure 6.1 Belt Friction experiment apparatus
36
6.3 Theory:
Belt drive machinery makes up significant portions of mechanical systems. Belt drive is used
in the transmission of power over comparatively long distances. In many cases, the use of
belt drive simplifies the design of a machine and substantially reduces the cost. Belt drive
employs friction for the transmission of power. The co-efficient of friction for belt drive
depends on the type of material used for the belt and the pulley.
The power (in Watt) transmitted by a belt is:
𝑃 = (𝑇2 − 𝑇1 )𝑉
Where 𝑉
(5.1)
is the velocity of the belt in meter per second; 𝑇2 is the initial tension on the
tight side
𝑇1 is the initial tension on the slack side.
The equation that relates the co-efficient of friction, tensions, the angle of wrap and the
angle of groove is:
𝜇𝜃
𝑇2
= 𝑒 ( ⁄sin 𝛼)
𝑇1
(5.2)
Where 𝑇2 is the initial tension on the tight side; 𝑇1 is the initial tension on the slack side ;
𝜇
is the co-efficient of friction; 𝛼
is the
total angle of groove in degrees (𝛼 =
90° 𝑓𝑜𝑟 𝑓𝑙𝑎𝑡 𝑏𝑒𝑙𝑡, 𝛼 = 20 ° 𝑓𝑜𝑟 𝑉𝑒𝑒 𝑏𝑒𝑙𝑡 ) . 𝜃 is the angle of wrap in radians measured from
the point of tangency of 𝑇1 and 𝑇2 .
6.4 Belt Friction 1: Flat Belt
6.4.1 Test Procedure
The procedure for the flat belt experiment is highlighted below:
1. Ensure that the apparatus is set-up as shown in Figure 6.1.
2. Unscrew the peg and place the angle marker at the 30 degree position and fasten
the peg.
3. Pass the flat belt over the pulley gently.
4. Place the weight (𝑇1 ) on the weight holder which is at the slack end of the belt.
37
5. Read and record the tensions on the tight side (spring balance reading) (𝑇2 ), and the
slack side (𝑇1 ). To read the tight side tension, rotate the pulley slowly and steadily
in a direction so that the spring balance side of the belt is in tension.
6. With increasing values of the weight (𝑇1 ), repeat steps (4) and (5) until five readings
are obtained and the corresponding (𝑇2 ) values are also obtained.
7. Repeat steps (4) to (6) with the angle marker at 60°, 90°, 120° and 150° .
6.4.2 Results and Discussions
Sample tables are presented in Table 6.1 to serve as a guide on how you are to present your
results. The results of the experiment should be presented for all peg angles. The peg angle
shown in Table 6-1 is for 30°.
Table 6-1 Sample data sheet for peg Angle 30°.
Peg Angle 30°
𝑇1 (𝑁)
𝑇2 (𝑁)
Experimental
Theoretical
𝑇2
𝑇1
𝑇2
𝑇1
Table 6-2 Sample data sheet for determination of co-efficient of friction
𝜃
Case study: 𝑇1 = ______𝑁
𝑇1 (𝑁)
𝑇2 (𝑁)
ln
𝑇2
𝑇1
Power
(𝑇2 − 𝑇1 )V
Complete the following Tasks:
1. Plot and discuss a graph of (𝑇2 ) versus (𝑇1 ) for all the peg angles (30° , 60°, 90°, 120°
and 150°) for the flat belt on one graph sheet.
38
a. Discuss the impact of the peg angle on the value of 𝑇2 .
b. Discuss the impact of 𝑇1 on the value of 𝑇2 .
2. Take 𝑇1 = _______𝑁 as the case study (see Table 6-2):
a.
𝑇
plot and discuss a graph of ln 𝑇2 against 𝜃.
1
𝑇
b. Determine the coefficient of friction. Since ln 𝑇2 =
1
𝜇
sin 𝛼
𝜃 is a straight line
relation should be obtained if the dynamic co-efficient of friction remains
constant throughout the experiment.
3. Assuming a velocity equals unity,
a. calculate the power that is transmitted at different peg angles for the case
study in question 2 above.
b. Plot and discuss a graph of Power transmitted versus the peg angle.
4. Name two other types of belt commonly used for belt drive.
5. Name some applications of Flat belt drive mechanisms.
6.5 Belt Friction 2: Vee-Belt
6.5.1 Test Procedure
The procedure for the flat belt experiment is highlighted below:
1. Ensure that the apparatus is set-up as shown in Figure 6.1.
2. Unscrew the peg and place the angle marker at the 30 degree position and fasten
the peg.
3. Pass the Vee belt over the pulley gently.
4. Place the weight (𝑇1 ) on the weight holder which is at the slack end of the belt.
5. Read and record the tensions on the tight side (spring balance reading) (𝑇2 ), and the
slack side (𝑇1 ). To read the tight side tension, rotate the pulley slowly and steadily
in a direction so that the spring balance side of the belt is in tension.
6. With increasing values of the weight (𝑇1 ), repeat steps (4) and (5) until five readings
are obtained and the corresponding (𝑇2 ) values are also obtained.
7. Repeat steps (4) to (6) with the angle marker at 60°, 90°, 120° and 150° .
39
6.5.2 Results and Discussions
Sample tables are presented in Table 6-3 to serve as a guide on how you are to present your results.
The results of the experiment should be presented for all peg angles. The peg angle shown in
Table 6-4 is for 30°.
Table 6-3 Sample data sheet for peg Angle 30°.
Peg Angle 30°
𝑇1 (𝑁)
𝑇2 (𝑁)
Experimental
Theoretical
𝑇2
𝑇1
𝑇2
𝑇1
Table 6-4 Sample data sheet for determination of co-efficient of friction
𝜃
(degree)
𝜃
(radians)
Case study: 𝑇1 = ______𝑁
𝑇1 (𝑁)
𝑇2 (𝑁)
ln
𝑇2
𝑇1
Power
(𝑇2 − 𝑇1 )V
30
60
90
120
150
Complete the following Tasks:
1. Plot and discuss a graph of (𝑇2 ) versus (𝑇1 ) for all the peg angles (30° , 60°, 90°, 120°
and 150°) for the Vee belt on one graph sheet.
a. Discuss the impact of the peg angle on the value of 𝑇2 .
b. Discuss the impact of 𝑇1 on the value of 𝑇2 .
2. Take 𝑇1 = _______𝑁 as the case study (see Table 6-2):
a.
𝑇
plot and discuss a graph of ln 𝑇2 against 𝜃 in radians.
1
𝑇
b. Determine the coefficient of friction. Since ln 𝑇2 =
1
𝜇
sin 𝛼
𝜃 is a straight-line
relation should be obtained if the dynamic co-efficient of friction remains
constant throughout the experiment.
40
3. Assuming a velocity equals unity,
a. calculate the power that is transmitted at different peg angles for the case
study in question 2 above.
b. Plot and discuss a graph of Power transmitted versus the peg angle.
41
7 FLYWHEEL
Equation Chapter (Next) Section 1
7.1 Introduction
A flywheel is a device for storing kinetic energy. 1It is a solid disc mounted on the shaft of
machines such as turbines, steam engines, diesel engines etc. When the load of such
machines suddenly increases or decreases, its function is to minimize the speed fluctuations
which occurs during the working of machines.
2
The Flywheel acquires kinetic energy from the machines. The capacity of storing of KE
(kinetic energy) depend on the rotational inertia of the flywheel. This rotational inertia is
called as Moment of Inertia of rotating object namely wheels. The moment of inertia of
body is defined as the measure of object’s resistance to the changes of its rotation.
The objective(s) of this experiment are to: (a) Determine the moment of inertia of a
flywheel about its own axis of rotation, and (b) Validate the theoretical calculations
experimental data.
7.2 Apparatus description
The apparatus is made up of a flywheel and an axle that passes through the centre of the
flywheel. The axle is supported with the aid of bearings. A line is marked on the flywheel’s
circumference so that the number of its revolutions can be easily counted. A cord is wound
on the axle, and it has a peg attached at its one and a load hanger on the other end. The
peg can enter into the axle of the apparatus after wrapping few turns of the cord on the
axle. The rotation of the flywheel is caused by the unwrapping of the cord when the mass
falls.
1
2
Source: https://blog.oureducation.in/moment-of-inertia-of-a-flywheel-by-falling-weight-method/
Source: https://blog.oureducation.in/moment-of-inertia-of-a-flywheel-by-falling-weight-method/
42
Figure 7.1 The flywheel experiment apparatus set-up3
7.3 Theory
The theory applied to determine the moment of inertia of the fly-wheel in this experiment,
is based on the work-energy equation [2]. If a mass (𝑚) attached to the flywheel shown in
Figure 7.2 is allowed to fall from height “h” , then the potential energy (𝑚𝑔ℎ) initially
stored by the mass is used to: (a) change the kinetic energy of the flywheel; (b) change the
kinetic energy of the mass itself; and (c) overcome the friction . This statement is expressed
mathematically in Eq. These relation is true at the instance when the mass (𝑚) gets
detached.
3
Source: C. G. Ozoegwu, Experiment Guide and reporting model: ME 315-Mechanical Engineering Laboratory I. Nsukka:
Unpublished, 2018.
43
Figure 7.2 Fly-wheel working principle: work-energy4
Mathematically,
𝑊𝑜𝑟𝑘 = 𝐾𝐸 𝑜𝑓 𝑚𝑎𝑠𝑠 + 𝐾𝐸 𝑜𝑓 𝑓𝑙𝑦𝑤ℎ𝑒𝑒𝑙 + 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑤𝑜𝑟𝑘
𝑚𝑔ℎ =
1
1
𝑚𝑉 2 + 𝐼𝜔2 + 𝑛𝑊𝑓
2
2
(6.1)
(6.2)
𝑊𝑓 is the friction work per turn; 𝐼 is the moment of inertia of the flywheel,
The flywheel makes N turns after the cord completely unwraps itself from the axle and
before it (the flywheel) comes to rest. This implies that
1 2
𝐼𝜔 = 𝑁𝑊𝑓
2
1 𝐼𝜔2
.
𝑁
This implies that 𝑊𝑓 = 2
(6.3)
Since 𝑉 = 𝑟𝜔. 𝑉 is the velocity of the mass assembly at the
point of hitting the table, 𝑟 is the radius of the axle, Substitution of thèse expressions into
Equation. 7.2 and making the moment of Inertia the subject of formular yields
𝐼=
𝑁𝑚 2𝑔ℎ
(
− 𝑟2)
𝑁 + 𝑛 𝜔2
(6.4)
𝑁 is the number of turns from the time the load hanger touches the table to when the
flywheel comes to rest.
4
Source: G. S. Sawhney, “Determination of Moment of Inertial of Flywheel,” in Mechanical Experiments and
Workshop Practice, New Delhi: I.K. International Publishing House Pvt. Ltd., 2009, pp. 28–30.
44
Angular velocity (𝜔) can be determined by counting the number of rotations of the flywheel
before it stops and the time (𝑡) that it takes to stop.
2𝜋𝑁
𝑡
(6.5)
𝑁𝑚
𝑔ℎ𝑡 2
( 2 2 − 𝑟2)
𝑁 + 𝑛 2𝜋 𝑁
(6.6)
𝜔=
𝐼𝑎 =
The height “h” can be determined by counting the number of axle rotation (n), it takes for
the flywheel before the mass hits the floor.
ℎ = 2𝜋𝑟. 𝑛
(6.7)
Substituting the expressions for 𝜔 and ℎ in Eq. 7.6 gives
𝑁𝑚
𝑔𝑛𝑡 2
𝐼𝑏 =
𝑟(
− 𝑟)
𝑁+𝑛
𝜋𝑁 2
(6.8)
7.4 Test Procedure (extracted from [1])
1. Adjust the length of the cord carefully so that the loop slips off the peg when the
weight-hanger just touches the table.
2. Hang slotted mass (m=7kg), heavy enough to rotate the flywheel if suspended.
3. Boldly mark the rim at the line most visible to you with the loaded hanger just
touching the table (alternatively, for this experiment, use the peg as the visual
indicator).
4. While holding the flywheel, add potential energy to the system by rotating it n times
so that the cord is wound round the axle n times. Avoid overlapping of the cord on
the axle.
5. Measure the height (h) of the hanger from the table.
6. Release the flywheel to make n rotations before the load hanger hits the table and
the cord slips off from the peg.
7. Start the stop watch at the instant the load hanger hits the table and measure the
time taken for the flywheel to come to rest.
45
8. Count the number of rotations (𝑁) made by the flywheel during this time.
9. Repeat the experiment seven times by changing the value of 𝑛.
7.5 Results
The readings should then be tabulated as follows; Calculate the moment of inertia for
each reading.
Table 7-1 Sample data sheet for the
𝑛
ℎ
𝑁
𝑡
𝐼𝑎
𝐼𝑏
1
2
Average values
Complete the following tasks:
1.
Calculate
the
moment
of
inertia
𝐼𝑎 and 𝐼𝑏 for each entry. Compare the average moment inertia for 𝐼𝑏 and
𝐼𝑎 .
2.
Plot and discuss a graph of 𝐼𝑏 and 𝐼𝑎 versus n.
3.
Discuss the sources of error.
4.
State some specific applications of a flywheel in a machine.
5.
Briefly discuss the work-energy conversion principle at work in a flywheel.
7.6 Discussions
46
8 WESTON DIFFERENTIAL PULLEY
Equation Chapter (Next) Section 1
8.1 Introduction
The main objective of this experiment is to determine the machine law and efficiency of a
Weston differential pulley.
8.2 Apparatus description
The differential pulley apparatus (see Figure 8.1) consists of two blocks A and B. The upper
block has two pulleys. The pulleys of the upper block turn together as one pulley. The lower
block carries a pulley in which the load hanger is attached. A closed chain links the upper
pulley block to the lower block. A spring balance, load hanger, some masses are also
provided with the apparatus.
Figure 8.1 Weston Differential Pulley Apparatus
47
8.3 Theory
A schematic representation of the Weston differential pulley block is shown on Figure 8.2.
The upper block A, has two pulleys (𝑃1 , 𝑃2 ,) with radius of R and r respectively. A load of
W is attached to the lower block B.
Figure 8.2 A schematic of a Weston differential Pulley Block
The displacement of the effort (𝑃) in one revolution of the upper-pulley block is 2𝜋𝑅. This
is equal to the length of the chain pulled over the larger pulley in the upper block A. Since
the smaller pulley (P2) in block A turns with the larger one, it therefore means that the
length of the chain released by the smaller pulley is 2𝜋𝑟.
The net-reduction in the chain length equals 2𝜋(𝑅 − 𝑟). This net shortening in the chain
length is divided into two equal portions of the chain supporting the load. It therefore means
that the upward distance covered by the load is 𝜋(𝑅 − 𝑟).
The velocity ratio becomes
𝑉. 𝑅 =
2𝜋𝑅
2R
=
𝜋(𝑅 − 𝑟) (𝑅 − 𝑟)
(7.1)
The mechanical advantage is
48
𝑀. 𝐴 =
𝑊
𝑃
(7.2)
The efficiency of the machine becomes
𝜂=
M. A (𝑅 − 𝑟) 𝑊
=
𝑉. 𝑅
2𝑅 𝑃
(7.3)
(𝑅 − 𝑟)
𝑊
2𝑅𝜂
(7.4)
Therefore,
𝑃=
A zero-intercept plot of 𝑃 against 𝑊 gives a straight line with slope 𝑆 =
efficiency 𝜂 =
(𝑅−𝑟)
2𝑅𝜂
such that the
(𝑅−𝑟)
.
2𝑅𝑆
8.4 Test Procedures
1. Place a load W on the load hanger. Note the initial position of the base of the load
hanger (𝑥𝑊1 ) and the initial position of the hook of the spring balance (𝑥𝐸1 ).
2.
Draw gradually on the spring balance until the load starts moving upward while the
effort moves downwards. At this point note the effort 𝑃 as read from the load
balance.
3.
Pull the spring through the maximum displacement and note the final position of
the base of the load hanger (𝑥𝑊2 ) and the final position of the hook of the spring
balance (𝑥𝐸2 ).
4. Record your results as in a data sheet similar to Table 8-1.
5. Repeat for all values on load 𝑊 on Table 8-1.
49
8.5 Results and Discussions
The readings should then be tabulated as follows;
Table 8-1 Sample data sheet for differential pulley block experiment
Data collected
S/No
𝑊
(kg)
𝑃
(kg)
𝑥𝑊1
(cm)
𝑥𝐸1
(cm)
Experimentally based
𝑥𝑊2
(cm)
𝑥𝐸2
(cm)
Effi.
(%)
MA
VR
Theoretically based
VR
Effi. (%)
1
2
3
4
5
6
Mean
Complete the following Tasks:
1. Graphically derive the machine law of the pulley.
2. What is the maximum Mechanical advantage and efficiency of the machine.
3. Generate a zero-intercept plot of 𝑃 against 𝑊 and estimate the efficiency from the
graph.
4.
Calculate the mean efficiency and compare with the efficiency determined
graphically.
8.6 Conclusions
50
9 GYROSCOPE
Equation Chapter (Next) Section 1
9.1 Introduction
The gyroscope apparatus is designed to enable a demonstration of the relationship in Eq. 9.1 which
relates the polar moment of inertia (𝐼) of a body rotating at an angular speed (𝜔), whose axis of
rotation is not fixed in space but itself rotates with an angular speed (𝜔𝑝 ) with the external torque
(𝑇) which must act on it to sustain the motion.
𝑇 = 𝐼 𝜔𝜔𝑝
(8.1)
The polar moment of inertia, also known as second polar moment of area, is a quantity used to
describe resistance to torsional deformation (deflection), in cylindrical objects (or segments of
cylindrical object) with an invariant cross-section and no significant warping or out-of-plane
deformation.
9.2 Apparatus
The apparatus consists of a disc mounted on the spindle of a small variable speed motor motor
carried in a trunnion frame which can be rotated by a second variable speed geared motor unit.
The rotor motor is carried in an aluminium ring mounted transversely in ball bearings in the trunnion
frame. Various masses may be attached to the motor
frame to balance the gyroscopic couple
present when the motor is running. The aluminium trunnion frame is mounted on the upper end of
the vertical shaft running in radial and thrust bearings. The lower end of this vertical shaft is
connected by a flexible coupling to the final drive shaft of the geared motor unit fixed to the base
plate of the instrument. To ensure that no force be transmitted to the vertical shaft bearing, a
counterweight equal to half what is used on the torque arm may be attached to the trunnion frame
The moment of inertia of the motor armature, shaft and disc may be accurately determined
experimentally by using the spare armature and disc supplied with the apparatus. This is done by
suspending the spare armature and disc on a fine wire attached to the cantilever suspension arm
which is a standard part of the apparatus.
51
Note: When precessing5 the position of balance is clearly determined by observing the position of
a slot on the torque arm relative to a broad white line on the frame. At the point of balance the
white line is completely covered.
Truunion frame
Load attached to
Balancing load
torque arm
Spin motor
Vertical shaft
Spare armature and disc
instrument
Precession motor
Base plate
Cantilever suspension arm
Figure 9.1 The gyroscope apparatus
The sped control unit (see Figure 9.2) houses the gyroscope (spin) and precessing rate controls,
the gyroscope rate recorder, a flashing lamp indicator giving one flash per revolution of the trunnion
frame and an electro-magnetic counter for recording the number of the revolution of the trunnion
frame against time. Two reversing switches allow the two motors to be run in clockwise or anticlockwise direction.
5
Precessing simply means the act of changing the orientation of the rotational axis of a rotating body.
52
Electromagnetic counter
Gyroscope rate recorder
Flashing lamp indicator
Gyroscope reverse switch
Precession reverse switch
Gyroscope (spin) rate
Precession rate control
control
Figure 9.2 Speed control unit
9.3 Theory
Let the disc in Figure 9.3 be spinning in a vertical plane with angular velocity 𝜔 and simultaneously
the axis of spin is rotating in a horizontal plane XOZ with an angular velocity 𝜔𝑝 . . Since the angular
momentum is a vector quantity and, applying the angular momentum of the disc may be
⃗⃗⃗⃗⃗ when in the position shown by the full line and vector ⃗⃗⃗⃗⃗
represented by vector 𝒐𝒂
𝒐𝒃 when in the
position shown by the broken lines. The change of angular momentum in the interval of time is
⃗⃗⃗⃗⃗ .
therefore the vector 𝒂𝒃
Figure 9.3 The spinning disc
The applied Torque (𝐼), that is responsible for the rate of change of the angular momentum, is
expressed mathematically as:
53
𝑇=
𝛿(𝐼𝜔)
𝛿𝑡
(8.2)
⃗⃗⃗⃗⃗ × 𝛿𝜃 , where 𝛿𝜃 is the angle through which the axis of spin rotates in the
But 𝛿(𝐼𝜔) = ⃗⃗⃗⃗⃗
𝒂𝒃 = 𝒐𝒂
time 𝛿𝑡. It therefore means that
𝛿𝜃
𝛿𝜃
= 𝐼𝜔
𝛿𝑡
𝛿𝑡
(8.3)
𝛿𝜃
= 𝐼. 𝜔. 𝜔𝑝
𝛿𝑡
(8.4)
⃗⃗⃗⃗⃗
𝑇 = 𝒐𝒂
In the limit, as 𝛿𝜃 → 0 then
𝑇 = 𝐼𝜔
𝑇 is the Torque, 𝐼 is the moment of inertia of the disc; 𝜔 is the angular velocity of thee disc in
𝑟𝑎𝑑/𝑠. 𝜔𝑝 is the angular velocity of precession in 𝑟𝑎𝑑/𝑠
Three experiments can be carried out to demonstrate the relationship in Equation (9.4). The
experiments are : (a) The constant torque test; (b) the constant precession rate test; and (c) the
Constant rotor speed test. The procedure for these experiments are discussed in the sub-sections
9.5 , 9.6, and 9.7 respectively.
9.4 Determination of Moment of Inertia of the Armature and Disc
Before carrying out any of the three aforementioned experiments, the moment of Inertia of the
spin (Gyroscope) motor armature and Disc can be determined experimentally by a torsional
pendulum6 experiment on a suspended armature and disc tied to a wire, length .
6
A torsional pendulum, or torsional oscillator, consists of a disk-like mass suspended from a thin rod or wire. When the
mass is twisted about the axis of the wire, the wire exerts a torque on the mass, tending to rotate it back to its original
position. If twisted and released, the mass will oscillate back and forth, executing simple harmonic motion. Source:
https://www.cmi.ac.in/~ravitej/lab/4-torpen.pdf
54
𝒍
Figure 9.4 The rotating mass (armature motor & Disc) suspended by a wire
The equation of motion for torsional oscillation of the suspended mass in Figure 9.4 is given by:
𝐼. 𝑙
𝑇𝑝𝑒𝑟𝑖𝑜𝑑 = 2𝜋√
𝐺𝐽
(8.5)
Rewriting it gives:
𝑇𝑝𝑒𝑟𝑖𝑜𝑑 2 𝐺𝐽
𝐼=
4𝜋 2 𝑙
(8.6)
Where 𝐼 is the moment of inertia of the rotor; 𝑇𝑝𝑒𝑟𝑖𝑜𝑑 is the period of oscillation, 𝐺 is the modulus
of rigidity of the wire; g is acceleration due to gravity; 𝐽 is the second moment of area about the
polar axis; length of the wire.
The second moment of area about the polar axis is defined as:
𝐽=
𝜋 𝑟4
2
(8.7)
𝑟 is the radius of the wire.
55
The procedure for completing this experiment is described below:
1. Suspend the extra armature motor & Disc with the aid of a string attached to the
Cantilever suspension arm.
2. Measure the length of the string.
3.
Give a small angular displacement to the system and leave it to oscillate.
4.
Then measure the time (using the electromagnetic counter), it takes to complete
between 7 and 10 oscillations
5. Repeat steps (c ) and (d) twice for the same wire length and then take the average time
so as to minimize the error due to our reaction time and precision of the pendulum.
The following data is provided to determine the moment of inertia:
•
•
•
The diameter of the string/wire is 0.7112mm.
The modulus of rigidity of the wire (𝐺) is 9.6526 × 1010 𝑁𝑚−2.
The acceleration due to gravity is 9.8 𝑚𝑠 −2 .
The sample data sheet for presenting the data generated from this experiment is shown on Table
9-1.
Table 9-1 Sample data sheet for determining the moment of inertia of the armature
No of Oscillations
Time taken (s)
Period for one
oscillation
1
2
3
Average period per oscillation
(𝑇𝑝𝑒𝑟𝑖𝑜𝑑 )
Use Equation (9.6) to determine the moment of inertia of the armature and Disc attached to the
string
56
9.5 Gyroscope 1 (Constant Torque (𝑻) Test)
9.5.1 Test Procedure
Ensure that you complete the procedure in section 9.4 before you commence with this
experiment. The procedure for completing this experiment are:
1. Measure (take) the torque arm of the load, and choose a constant load (𝑊).
2. In clockwise mode, set the precession motor rotating at a low (timed) speed, and adjust
the speed of the spin motor (also in clockwise mode) until the gyroscopic torque is balanced
by 𝑊, i.e until the spin motor frame floats freely. Note: the speeds of the precession motor
and the spin (gyroscope) motor.
3. Increase the precession speed in steps and repeat the experiment for four more readings.
(i.e you need five readings in total). Note: the results obtained so far are for “increasing
speed mode”
4. Obtain similar results for “decreasing speed mode” by stopping the system to a standstill,
then reversing the rotation of both spin (Gyroscope) and precession motors and repeating
the tests as the speeds are reduced in steps.
9.5.2 The results and Discussion
A sample data sheet for presenting the results for the constant Torque test in the clockwise mode
is shown on Table 9-2. A similar table should be made for the counter-clockwise operation as
well.
57
Table 9-2 Sample data sheet for constant torque test in clockwise mode
Constant Load =
Constant Torque =
Mode : Precession clockwise and Rotor Spinning clockwise
Readings No
1
2
3
4
5
Rotor (spin) speed in (𝑅𝑒𝑣/𝑚𝑖𝑛)
Rotor (spin) speed ( 𝜔 ) (𝑅𝑎𝑑/
𝑠)
Precession speed in (𝑅𝑒𝑣/𝑚𝑖𝑛)
Precession speed (𝜔𝑝 ) (𝑅𝑎𝑑/𝑠)
Complete the following Tasks:
1. Plot and discuss the graph of (1/𝜔𝑝 ) against 𝜔 for the clockwise operation. Draw a line of
best fit across the plot.
2. Plot and discuss the graph of (1/𝜔𝑝 ) against 𝜔 for the anti-clockwise operation. Draw a
line of best fit across the plot.
3. What inference can you make from both plots.
4. Determine the moment of inertia in each of the plots, and take an average of both.
5. Compare this average moment of inertia with the one determined in section 9.4.
6. Discuss any source of discrepancy in the data.
9.6 Gyroscope 2 (Constant Precession rate (𝝎𝒑 ) Test)
9.6.1 Test Procedure
Ensure that you complete the procedure in section 9.4 before you commence with this
experiment. The procedure for completing this experiment are:
1. In clockwise mode, keep the precession speed constant at about
6.275 rad/s = 60
Rev/min.
2. For each load (𝑊) attached to the motor frame, determine the rotor speed (𝜔) needed
for balance.
3. After carrying the experiment for increasing 𝜔, reverse the rotation of both motors and
repeat for decreasing, 𝜔. Remember to bring the system to a standstill, before reversing
58
the rotation of both spin (Gyroscope) and precession motors and repeating the tests as
the speeds are reduced in steps.
9.6.2 Results and Discussion
A sample data sheet for presenting the results for the constant Precession test in the clockwise
mode is shown on Table 9-3. A similar table should be made for the counter-clockwise operation
as well.
Table 9-3 Sample data sheet for constant Precession speed test in clockwise mode
Constant Precession speed (rev/min) =
Constant Precession speed (𝜔𝑝 ) (rad/s) =
Mode of operation: Precession clockwise and Rotor Spinning clockwise
Readin
gs
Load
(N)
Actual
Torque
(Nm)
(𝑇)
Rotor
Speed
(Rev/min
)
Rotor
Speed
𝜔
(Rad/s)
Calculated
Torque*
% 𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑇𝑜𝑟𝑞𝑢𝑒
Calc. Torque − Act Torque
=|
| 100
Act Torque
1
2
3
4
5
* The calculated Torque is done 𝑇 = 𝐼. 𝜔. 𝜔𝑝 Use the moment of inertia 𝐼 determined from
section 9.4.
59
Complete the following Tasks:
1. Plot and discuss the graph of (𝑇) against 𝜔 for the clockwise operation. Draw a line of
best fit across the plot
2. Plot and discuss the graph of (𝑇) against 𝜔 for the anticlockwise operation. Draw a line of
best fit across the plot
3. What inference can you make from both plots.
4. Determine the moment of inertia in each of the plots, and take an average of both.
5. Compare this average moment of inertia with the one determined in section 9.4.
6. Discuss any source of discrepancy in the data.
9.7 Gyroscope 3 (The constant rotor speed (𝝎) Test)
9.7.1 Test Procedure
The procedure for completing the experiment are:
1. In clockwise mode, keep the sin motor rotor speed constant at about 6.275 rad/s = 60
cycles/min.
2. For each load (W) attached to the motor frame, determine the rotor speed (𝜔𝑝 ) needed for
balance.
3. After carrying the experiment for increasing (𝜔𝑝 ) , reverse the rotation of both motors and
repeat for decreasing, (𝜔𝑝 ) . Remember to bring the system to a standstill, before reversing
the rotation of both spin (Gyroscope) and precession motors and repeating the tests as the
speeds are reduced in steps.
9.7.2 Results and Discussion
A sample data sheet for presenting the results for the constant Precession test in the clockwise
mode is shown on Figure 9.4. A similar table should be made for the counter-clockwise operation
as well.
60
Table 9-4 Sample data sheet for constant rotor speed test in clockwise mode
Constant Rotor speed (rev/min) =
Constant Rotor speed 𝜔 (rad/s) =
Mode of operation: Precession clockwise and rotor spinning clockwise
Reading
s
Load
(N)
Actual
Torque
(Nm)
(𝑇)
Precession
Speed
(Rev/min)
Precession
Speed
(𝜔𝑝 )
(Rad/s)
Calculated
Torque*
% 𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑇𝑜𝑟𝑞𝑢𝑒
Calc. Torque − Act Torque
=|
| 100
Act Torque
1
2
3
4
5
* The calculated Torque is done using 𝑇 = 𝐼. 𝜔. 𝜔𝑝 Use the moment of inertia 𝐼 determined from section
9.4.
Complete the following Tasks:
1. Plot and discuss the graph of (𝑇) against 𝜔𝑝 for the clockwise operation. Draw a line of
best fit across the plot
2. Plot and discuss the graph of (𝑇) against 𝜔 for the anticlockwise operation. Draw a line of
best fit across the plot
3. What inference can you make from both plots.
4. Determine the moment of inertia in each of the plots, and take an average of both.
5. Compare this average moment of inertia with the one determined in section 9.4.
6. Discuss any source of discrepancy in the data.
61
10 CENTRIFUGAL FORCE
Equation Chapter (Next) Section 1
10.1 Introduction and Theory
A body moving in a circular path of radius r, with an angular velocity ω, is continuously accelerated
towards the centre of the circular path by an acceleration called the centripetal or centre seeking
acceleration.
It follows from Newton's second law of motion, that a force must act on the mass m, in the direction
of this acceleration i.e., a centripetal force of magnitude mω2r. The reaction provided to
counteract this centripetal force is in the opposite sense, i.e acting outwards, and it is called the
Centrifugal force.
Figure 10.1 Schematic diagram of the Centrifugal Apparatus
The bell crank is pivoted at O and revolving about Y with angular velocity ω. Let the masses at A
and B be 𝑀𝑎 and 𝑀𝑏 respectively and at corresponding radii a and b. Considering the rotation of
the bell crank about Y, the line of action of the centrifugal force due to mass B passes through
the pivot O, and therefore has no moment about O, since the arm OA is vertical. Taking moment
about zero;
62
𝐹𝑐 = 𝑀𝑎 𝜔2 𝑟𝑎 = 𝑀𝑏 𝑏𝑔
𝜔2 =
16.817𝑀𝑏
𝑟. 𝑀𝑎
(9.1)
(9.2)
At this angular velocity ω, the centrifugal force is equal to the force due to 𝑀𝑏 .
The objective of this experiment is to analyse centrifugal Force using Centrifugal Force apparatus
bell crank.
10.2 Apparatus
Figure 10.2 shows the centrifugal force apparatus. It has a large metallic container that is placed
on a rotating base. The metallic container has provisions for coupling various masses.
Figure 10.2 Centrifugal Apparatus
63
10.3 Centrifugal force 1 (varying Ma, constant Mb & r)
To verify the centrifugal force equation, 𝐹 = 𝑚𝜔2 𝑟 by establishing the relationship between ω2
and 1/𝑀𝑎 .
10.3.1 Test Procedure
Put on the apparatus and allow it to run for a few minutes before commencing the tests. Stop the
motor when you are ready to perform the tests.
1. Attach two masses (𝑀𝑏 ) of equal magnitude to the horizontal arms of the apparatus.
2. Attach two masses (𝑀𝑎 ) of equal magnitude to the vertical arms of the apparatus.
3. Lock the masses to the bell brackets with their lock nuts.
4. Lock the two brackets C at equal distance r from the centre Y, with the locating pins.
5. Start the motor and very slowly and evenly increase the speed until the two bell-cranks snap
over from stop D to stop E. You will hear a click. At this precise moment, leave the speed
control stationary.
6. Simultaneously start a stopwatch and the magnetic revolution counter on the control box.
With these, we will be able to calculate the angular speed of the rotating arm.
7. After a time, lapse of 20 secs, stop the stopwatch and the revolution counter simultaneously
and read the time duration t, and number of revolutions.
8. With the mass 𝑀𝑏 , and distance 𝑟, constant, change the value of 𝑀𝑎 , and repeat 2 - 7.
10.3.2 Results and Discussions
Complete Table 10-1 using the data obtained from the experiment and the theoretical results
obtained.
Table 10-1 Sample data sheet for Centrifugal force 1 experiment
Mb =
ω2
;r=
Ma
Counter
Time
Theoretical
Practical
1/𝑀𝑎
64
Complete the following tasks:
i.
Plot and discuss a graph of the theoretical and experimental ω2 against
ii.
What are some of the safety precautions that you observed?
1
𝑀𝑎
10.4 Centrifugal force 2 (varying Mb, constant Ma & r)
To verify the centrifugal force equation, F = mω2r by establishing the relationship between ω2 and
F
10.4.1 Test Procedure
Put on the apparatus and allow it to run for a few minutes before commencing the tests. Stop the
motor when you are ready to perform the tests.
1. Attach two masses (𝑀𝑏 ) of equal magnitude to the horizontal arms of the apparatus.
2. Attach two masses (𝑀𝑎 ) of equal magnitude to the vertical arms of the apparatus.
3. Lock the masses to the bell brackets with their lock nuts.
4. Lock the two brackets C at equal distance r from the centre Y, with the locating pins.
5. Start the motor and very slowly and evenly increase the speed until the two bell-cranks snap
over from stop D to stop E. You will hear a click. At this precise moment, leave the speed
control stationary.
6. Simultaneously start a stopwatch and the magnetic revolution counter on the control box.
With these, we will be able to calculate the angular speed of the rotating arm.
7. After a time lapse of 20secs, stop the stopwatch and the revolution counter simultaneously
and read the time duration t, and number of revolutions.
8. With the mass Ma, and distance r, constant, change the value of 𝑀𝑏 , and repeat 2 - 7.
10.4.2 Results and Discussions
Complete Table 10-2 using the data obtained from the experiment and the theoretical results
obtained.
65
Table 10-2 Data Sheet for Centrifugal force 2 Experiment
Ma =
ω2
;r=
Mb
Counter
Time
Theoretical
Practical
1.714Mbg
Complete the following Tasks:
(1.) Plot and discuss a graph of the theoretical and experimental ω2 against 1.714 ∙ 𝑀𝑏 ∙ 𝑔
10.5 Centrifugal force 3 (varying r, Constant Mb & Ma)
10.5.1 Test Procedure
Put on the apparatus and allow it to run for a few minutes before commencing the tests. Stop the
motor when you are ready to perform the tests.
1. Attach two masses (𝑀𝑏 ) of equal magnitude to the horizontal arms of the apparatus.
2. Attach two masses (𝑀𝑎 ) of equal magnitude to the vertical arms of the apparatus.
3. Lock the masses to the bell brackets with their lock nuts.
4. Lock the two brackets C at equal distance r from the centre Y, with the locating pins.
5. Start the motor and very slowly and evenly increase the speed until the two bell-cranks snap
over from stop D to stop E. You will hear a click. At this precise moment, leave the speed
control stationary.
6. Simultaneously start a stopwatch and the magnetic revolution counter on the control box.
With these, we will be able to calculate the angular speed of the rotating arm.
7. After a time lapse of 20 secs, stop the stopwatch and the revolution counter simultaneously
and read the time duration t, and number of revolutions.
8. With the mass 𝑀𝑏 , and mass 𝑀𝑎 , constant, change the value of the distance r by moving
the bell bracket to the next hole on the arm; and repeat 2 - 7.
66
10.5.2 Results and Discussions
Complete Table 10-3 using the data obtained from the experiment and the theoretical results
obtained.
Table 10-3 Data Sheet for Centrifugal force Experiment 3
Ma =
ω2
; Mb =
r
Counter
Time
Theoretical
Practical
1/r
1. Plot and discuss the graph pf the theoretical and experimental ω2 against 1/𝑟
67
11 TORSION OF ROD
Equation Chapter (Next) Section 1
11.1 Introduction
Torsion occurs when any shaft is subjected to a torque. This is true whether the shaft is
rotating
(such as drive shafts on engines, motors and turbines) or stationary (such as with a bolt or
screw). The torque makes the shaft twist and one end rotates relative to the other inducing
shear stress on any cross section.
The objective of this experiment is to ascertain the modulus of rigidity of a rod by statical
method, using horizontal pattern of torsion test apparatus (Clamp pattern type). Modulus
of rigidity (G) is defined as the shear stress divided by the shear strain.
One very important use of circular shafts is for the transmission of power. Shafts have to
bear torsion, bending and axial forces. The shaft is said to be under pure torsion if there
are no bending and axial forces acting on it.
11.2 Apparatus
The apparatus for this experiment is shown on Figure 11.1. It has (i) grips to hold rod, (ii)
left grip having pulley and other end with a load hanger (pan) attached, (iv) weights to be
kept on the pan, (v) twist measuring circular calibrations provided on the pulley, and (vi)
Two indicators moving on the circular calibration provided on the apparatus. The specimen
rod is subjected to a torque force based on the load attached to the pulley. The specimen
is attached to pulley through grip twists when torque is applied which is read by the position
of the indicator.
68
Scale for reading 𝜃1
Specimen rod
Scale for reading 𝜃2
Pulley with cord wrapped
Gauge length
Figure 11.1 Torsion testing apparatus
Some of the other instruments required include: A Vernier calliper or micrometre & steel
rule.
11.3 Theory
Under pure torsion load conditions, the cross-section of a shaft is under pure shear stress
only. The shaft undergoes deformation by an angle 𝜃𝑟 (in radians) due to the applied torque.
The relationships between (i) torque (𝑇); (ii) radius and shaft ( 𝑟 ); (iii) length of the shaft
(𝑙); (iv) maximum shear stress transmitted to the material of the shaft (𝜏𝑠 ) ; (v) Twist angle
(𝜃𝑟 ) in radians and (vi) modulus of rigidity (𝐺) is given by the torsion equation as:
𝑇 𝐺𝜃𝑟
=
𝐼𝑝
𝑙
𝐼𝑝 is the polar moment of inertia, and it is equal to
𝑇=
(10.1)
𝜋𝑟 4
.
2
𝑟 is the radius of the rod.
𝑚. 𝑔. 𝐷
2
(10.2)
𝑚 is the mass of the weight on the pan. 𝐷 is the diameter of the pulley.
If the angle of twist is measured in degree, then it can be converted to radians (𝜃𝑟 ).
69
11.4 Torsion 1 Experiment
11.4.1 Test procedure
In this experiment, the gauge length of the shaft is kept constant while the relative angular
deflection or twist angle measurements are taken at various loads.
The procedure for the experiment are as follow:
1. Measure the gauge length of the specimen rod.
2. Clamp the rod and measure the length of the rod between the jaws
3. Adjust the indicator pointer on the two circular scales provided on the apparatus,
to read zero.
4. Place weight on the pan and read the twist angle (𝜃1 ) on the first scale and (𝜃2 )
on the second scale through which the specimen rod twists.
5. Take nine extra data points by increasing the weights in the pan and note down
the corresponding twists in the specimen rod.
Note:
•
•
•
Maximum length of the shaft specimen is 45cm
Diameter of specimen is 4.75mm
Effective radius of cord pulley is 10 cm
70
11.4.2 Results and discussions
A sample data sheet used for reporting this experiment is shown on
Table 11-1.
Table 11-1 Sample data sheet for Torsion of Rod
S/No
Load on
pan (𝑊)
In
Newtons
Twist
Twist
angle on
angle on
the first
the second
scale (𝜃1 ) scale (𝜃2 )
in degree
in degree
Constant gauge length used (𝑙) = ___________
Average
Twist
angle
(𝜃𝑎𝑣 ) in
degree
cm
Average
Twist angle
(𝜃𝑟 ) in
radians
Torque
(𝑇) in
Nm
Calculated
modulus of
rigidity (𝐺)
using Eq.
11.1
% error
1
2
3
4
5
6
7
8
9
10
Complete the following Tasks:
1. Plot and discuss the graph of the angular twist
(𝜃𝑟 ) in radians against torque (𝑇)
2. Plot and discuss the graph of the first angular twist
(𝜃1 ) and the second angular
twist (𝜃2 ) on the y-axis, versus the Torque (𝑇) on the x-axis.
3. Determine the modulus of rigidity from the plot drawn in (a) above.
4. Calculate the average theoretical modulus of rigidity of the specimen rod using
equation (19.4). This result should match the one obtained for the specimen on
the data sheet.
71
11.5 Torsion 2
11.5.1 Test Procedure
In this experiment, the applied load is kept constant while the angular deflection / twist
angle readings are taken for various gauge length. The effective gauge length is the distance
from the centre of the clamp of one pointer (scale) to the centre of the clamp of the second
pointer (scale).
The procedure for the experiment are as follow:
1. Measure the diameter of the specimen rod
2. Measure the diameter of the pulley.
3. Select a constant load that would be used throughout this experiment.
4. Clamp the rod and measure the length of the rod between the jaws
5. Adjust the indicator pointer on the circular scale to read zero
6. Place weight on the pan and read the angle (𝜃) through which the specimen rod
twists.
7. Adjust the position of the specimen rod, so that the effective gauge length is
reduced.
8. Take nine extra data points by increasing the gauge length of the specimen rod, so
that the effective gauge length is reduced.
Note:
•
•
•
Maximum length of the shaft specimen is 45cm
Diameter of specimen is 4.75mm
Effective radius of cord pulley is 10 cm
72
11.5.2 Results and discussions
A sample data sheet used for reporting this experiment is shown on
Table 11-1.
Table 11-2 Sample data sheet for Torsion of Rod
S/No
Load on
pan
(𝑊)
Twist angle
on the first
scale (𝜃1 ) in
degree
Twist angle
on the
second scale
(𝜃2 ) in
degree
Average
Twist
angle
(𝜃𝑎𝑣 ) in
degree
Average
Twist
angle
(𝜃𝑟 ) in
radians
Gauge
length
(𝑐𝑚)
Calculated
modulus of
rigidity (𝐺)
using
Equation
11.1
%
error
Constant load (𝑊) = __________ (N)
Constant Torque (𝑇) = _________(Nm)
1
2
3
4
5
6
7
8
9
10
Complete the following Tasks:
1. Plot and discuss the graph of the angular twist
(𝜃𝑟 ) in radians against gauge length
(𝑙)
2. Plot and discuss the graph of the first angular twist
(𝜃1 ) and the second angular
twist (𝜃2 ) on the y-axis, versus the gauge length (𝑙) on the x-axis.
3. Determine the modulus of rigidity from the plot drawn in (a) above.
4. Calculate the average theoretical modulus of rigidity of the specimen rod using
equation (11.1). This result should match the one obtained for the specimen on the
data sheet.
5. State the safety precautions.
73
12 IMPACT TESTING
Equation Chapter (Next) Section 1
12.1 Introduction
To determine the impact strength of a specimen by (a) Charpy method and (b) Izod method
using an impact testing machine.
12.2 Apparatus
The impact testing machine (see Figure 12.1) consists of a swinging mass or pendulum7
(294.212 Joules) (217 ft-lbs). Which is made to strike a sudden blow on the specimen on the
specimen. The pendulum is made to swing from a specified height and it hits the specimen.
After striking and breaking the specimen, the height to which the pendulum rises depends
upon the amount of energy that is absorbed by the specimen during rupture. The height to
which the pendulum rises after striking the specimen, is indirectly proportional to the
energy absorbed by the specimen during the test. Three different scales are provided on
the apparatus (see Table 12-1):
Table 12-1 Scales on the Impact testing Apparatus
Scale position
Quantity measured
Unit of scale provided on the
instrument
Inside scale
The notch specimen toughness or
𝑘𝑔𝑓𝑚/𝑐𝑚2 or 𝑘𝑝𝑚/𝑐𝑚2
impact strength
Central scale
Consumed or fracture or impact
𝑓𝑡 − 𝑙𝑏𝑠
(energy)
Outside scale
Angle subtended by the pendulum
Degree (°)
after the first swing
Note on the Instrument Scale:
•
𝑘𝑔𝑓𝑚/𝑐𝑚2 is kilogram-force per square centimetre square. This is a unit of stress or
pressure. 1 𝑘𝑔𝑓𝑚/𝑐𝑚2 is equal to98.07 𝑘𝑁/𝑚2 .
•
𝑓𝑡 − 𝑙𝑏𝑠 is foot-pound. It is a unit of energy. 1 𝑓𝑡 − 𝑙𝑏𝑠 is equal to 1.35582 𝐽𝑜𝑢𝑙𝑒𝑠.
7
The hammer with additional weights is 217 ft-lbs (294.212 Joules) while the hammer without additional
weight 108.5 ft-lbs (147.106 Joules).
74
There are two tests which can be conducted using the apparatus: (a) The Charpy impact
test; and (b) The Izod impact test. The main difference between the two types of tests is
the position in which the specimen is held (see Figure 12.2).
Figure 12.1 Schematic diagram of an impact testing machine
Figure 12.2 Difference between Charpy and Izod Impact test
In the Charpy test, the specimen is held horizontally as a simply supported beam in the anvil
of the machine. However, in the Izod test, the specimen is held vertically as a cantilever in
the anvil of the machine. The specimen in both the test has a V-shaped notch (45 degree
angle included) and the depth of the notch is generally kept between one fifth (1/5) and
one third (1/3) of the thickness of the specimen. However, in Charpy test, the notch is kept
75
at the opposite side of the specimen facing the blow while it is kept in the facing side of
the specimen in the Izod test.
12.3 Theory
The ability of a material to withstand impact of shock loading is referred to as toughness.
This is determined by measuring the amount of energy which a material can absorb before
it fails. Toughness is a combined effect of both strength and ductility. The impact test
simulates similar service conditions which are often encountered by transport, agricultural
and construction equipment. The stresses induced during impact loading are more than the
ones induced during sudden/gradual loading.
Typically, a notch is introduced to the specimen, so that it fails at the notch when it is
subjected to sudden impact loading. A swinging mass is given a potential energy by swinging
it from a height which breaks the specimen by impact loading. This energy for fracture can
be measured by finding the difference of height gained after rupture from initial height by
the swinging mass. i.e. the total energy absorbed by the specimen can be inferred from the
height difference of the pendulum (angle subtended by the pendulum). Hence it is given as:
𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 = 𝑚𝑔(𝐻 − ℎ)
(11.1)
76
First position
First reversal point
Figure 12.3 Principle of Impact testing using height measurements
Alternatively, if the angle of fall (𝛼), and the angle at the end of the first swing (𝛽) is
known, then the impact energy absorbed by a material can be calculated using the
equation (12.2):
Figure 12.4 Principle of Impact testing using angle of fall8 and rise measurements
8
Source: https://saecanet.com/calculation_page/000380_000509_Charpy_impact_test.php
77
𝐸𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 (𝐽) = 𝑊𝑔𝑅 (cos 𝛽 − cos 𝛼) − 𝐿𝑙𝑜𝑠𝑠𝑒𝑠
(11.2)
𝑊 is the mass of the hammer, which is equal to 19.36 kg. 𝑔 = 9.806𝑚/𝑠2 ; 𝑅 is the
length between centre of pivot point and gravity centre of hammer. 𝛼 is the angle of fall
which is a fixed value if all tests start with the pendulum hammer suspended in the
specified position. 𝛽 is the angle of rise during the first swing. 𝐿𝑙𝑜𝑠𝑠𝑒𝑠 is the losses especially
due to friction losses.
12.4 Test Procedure
You would be provided with test samples of different materials. The following steps should
be taken to complete the experiment for each material type provided:
1. Lift up the pendulum and locked it while suspended up. Measure
a. The height of the pendulum bulb head from the ground. This is the
ℎ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 .
b. The length between centre of pivot point and gravity centre of
hammer(𝑅).
c. The typical angle of fall (𝛼) when there is no specimen on the anvil. Do
this by gently dropping the pendulum hammer to the lowest position to
the ground. Then measure the angle on the scale.
2. Ensure that the pointer is at the zero reading
3. Ensure that the anvil is properly positioned
4. Mount the specimen in the anvil of the machine when the pendulum is locked
at the top position. Ensure that the notched end is facing inward.
5. Release the pendulum so that it breaks the specimen
6. Apply the break handle gradually until the swinging of the pendulum stops.
7. Note down the three readings of the pointer on the scales provided.
8. Repeat steps (a) to (g) t the experiment thrice for e
9. Repeat the experiment for other specimens.
Note: You would be provided with specimens made from different materials.
78
12.5 Results and Discussions
The test results for a given material can be tabulated as shown on Table 12-2.
Table 12-2 Standard data sheet for reporting Impact test of a given material
Material of the specimen =
Size of the specimen = length X breadth X Width
Initial reading measured
(ℎ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ) in meters =
𝑅 is the length between center of pivot point and gravity center of hammer =
Measure typical angle of fall (𝛼) in degree =
Specimen Number
1
2
3
4
5
losses
The notch specimen
toughness or impact strength
(Inside scale) measure in
𝑘𝑔𝑓𝑚/𝑐𝑚2 or 𝑘𝑝𝑚/𝑐𝑚2
Consumed or fracture or
impact (energy) (Central
scale) in measured in 𝑓𝑡 −
𝑙𝑏𝑠
Angle of rise of the pendulum
after the first impact ( 𝛽)
measured in degree
Angle of fall before impact 𝛼
The notch specimen
toughness or impact strength
(Inside scale) conversion to
𝑘𝑁/𝑚2
Consumed impact (energy)
(Central scale) conversion to
𝐽𝑜𝑢𝑙𝑒𝑠.
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝐼𝑚𝑝𝑎𝑐𝑡 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 (𝐽) =
𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 + 𝐸5
5
the impact energy absorbed
(𝐸) is calculated using
Equation 12.2 )
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐼𝑚𝑝𝑎𝑐𝑡 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 (𝐽) =
𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 + 𝐸5
5
Note: The fracture energy is determined from the swing-up angle of the hammer and its
swing-down angle. The Charpy impact value (kJ/m2) is calculated by dividing the fracture
energy by the cross-section area of the specimen.
Complete the following tasks:
79
1. Plot and discuss a single bar chart showing the consumed energy (in Joules) for
each test conducted for all the materials. The chart should look similar to the
one shown on Figure 12.5.
2. Plot and discuss a single bar chart showing the angle (𝜃) subtended by the
pendulum for each test conducted for all the materials. The chart should look
similar to the one shown on Figure 12.5.
3. Compare the theoretical impact energy absorbed and the experimental impact
energy absorbed for each specimen.
4. State the reason(s) why the theoretical value is different from the experimental
value.
5. Rank the test materials types in ascending order of impact strength.
6. State the safety precautions
Ferrous Materials
Test 5
Test 4
Test 3
Test 2
Test 1
Material type
Sample B
Average energy absorbed for material type
Test 5
Test 4
Test 3
Test 2
Test 1
Sample A
0
10
Average energy absorbed for material type A
20
30
40
50
60
70
Consumed impact energy (Joules)
Figure 12.5 A Sample of the bar chart for impact testing results
12.6 Conclusions
80
13 HARDNESS TESTING
Equation Chapter (Next) Section 1
13.1 Introduction
Hardness is a property of a material by which it resists any penetration under pressure,
scraping by a sharp point or abrasion when one metallic surface is rubbed against another
surface. In many Engineering applications, two surfaces in contact are required to roll or
slide over each other. These rolling or sliding surfaces are likely to develop scratch and
wearing down which weakens the surfaces leading to failure of the surfaces.
The hardness of a material is generally determined by measuring material resistance to
indentation. The indentor is usually a hardened ball. During the test, a load is applied by
pressing the indentor at right angle to the surface being tested.
There are different methods for measuring the hardness of a material. The four most
common methods are: (a) The Rockwell hardness test; (b) The Brinell Hardness test; (c )
The Vickers hardness test; ( D) The Knoop’s hardness test.
The hardness test in this course would be implemented by using the Rockwell hardness test.
This method’s commercial popularity arises from its speed, reliability, robustness,
resolution and small area of indentation .
13.2 Apparatus
The apparatus used for the Rockwell hardness test is shown on Figure 13.1. The main
components of the instrument are: (a) A dial which has two scales (B & C) for measuring the
depth of the indentation; (b) The penetrator or indenter which could be made of a diamond;
(c) The anvil/workbench which is used to carry the test piece; (d) the hand wheel which is
used to move the test sample up and down; (e) and the Crank release /load selector for
choosing the major load used in the test.
81
Dial face
Indentor
Specimen goes on
anvil
Hand
Wheel
Indentor
Crank to release
major load
Figure 13.1Rockwell hardness testing machine9
13.3 Theory
The determination of the Rockwell hardness of a material involves (see Figure 13.2 ) the
application of a minor load (pre-load) followed by a major load (Pre-load + Additional load). The
minor load establishes the zero position. The major load is applied, then removed while still
maintaining the minor load. The depth of penetration (ℎ) from the zero datum is measured from
a dial, on which a harder material gives a higher hardness number. That is, the penetration
depth (ℎ) and hardness are inversely proportional.
9
Source:
https://www.google.com/search?q=description+of+a+rockwell+hardness+apparatus&sxsrf=ACYBGNQAbWqy5
XKX63rlDh62x4GwJTs8nw:1579230799713&tbm=isch&source=iu&ictx=1&fir=My0vxPdaTeinWM%253A%252C
1XODfmRHKFYbHM%252C_&vet=1&usg=AI4_kQ0RSgwsmLYG7Oes37UfbXhpuwq1g&sa=X&ved=2ahUKEwjp9taW1YnnAhXQN8AKHb0RC2kQ9QEwDnoECAk
QBQ#imgrc=My0vxPdaTeinWM:
82
Loading Stage 1.
Pre-load
Loading Stage 2.
Pre-load + additional
load
h0
Loading Stage 3. Additional load
is removed. Measurement is
taken at this stage
hmax
h
Note: hmax is the maximum indentation depth. h is the lasting impression depth. h0 is the indention
depth by the pre-load.
Figure 13.2 Figure Force-depth diagram of Rockwell test10
The equation for Rockwell Hardness number calculated thus:
𝐻𝑅 = 𝑁 −
ℎ
𝑠
(12.1)
where ℎ is the depth (from the zero load point), and 𝑁 and 𝑠 are scale factors that depend
on the scale of the test being used. The different scales of Test that could be done are
shown on Table 13-1. Scale B and C are the most common ones used. Both Scales express
the hardness as an arbitrary dimensionless number. For instance, if the Rockwell hardness
number of a material is given as HRC 30, it simply means that the Rockwell hardness number
of the material is 30, and that it was measured using a Rockwell hardness Scale of C.
10
Source: https://www.emcotest.com/en/the-world-of-hardness-testing/hardness-know-how/theory-ofhardness-testing/rockwell/rockwell-test-procedure/
83
Table 13-1 Various Rockwell Hardness Scales11
13.4 Test Procedure
Test samples from different materials have been provided for the test. The procedure for
the Rockwell hardness testing on each test sample provided is stated below:
(1) Sample preparation: Ensure that the surface of all the test samples provided is
smooth.
(2) Place a test sample on the indenting table.
(3) Select the required load (in kilograms) needed for testing the sample.
a. 60 kilograms for Polymers,
b. 100 kilograms for non-ferrous metals like Aluminium, Copper and Brass,
c. 150 kilograms for ferrous metals such as mild steel and cast Iron.
(4) Turn the spin clockwise to move the sample close to the diamond indenter and stop
as soon as the sample makes contact with the diamond indenter.
(5) Select the appropriate scale for the sample from the dial. The scales are:
a.
11
Scale B for non-Ferrous metals ( shown in colour red)
Source: https://en.wikipedia.org/wiki/Rockwell_scale
84
b. Scale C for Ferrous metals (shown in colour black), and
(6) Set the pointer on the zero mark and turn it slowly till the pointer moves over the
zero mark
(7) Load an initial force (pre-load).
(8) Load the main load, and leave the main load for a "dwell time" sufficient for
indentation to come to a halt.
(9) Release load; the Rockwell value will typically display on a dial or screen
automatically.
(10)
Remove the indenter from the specimen
(11)
Repeat steps (4) – (9) at four different points on the test sample.
In order to get a reliable reading the thickness of the test-piece should be at least 10
times the depth of the indentation. Also, readings should be taken from a flat
perpendicular surface, because convex surfaces give lower readings. A correction factor
can be used if the hardness of a convex surface is to be measured.
13.5 Results and Discussions
A sample data sheet for presenting the test result for each test sample material is shown
on Table 13-2. A similar table should be prepared for each material type.
Table 13-2 Data Sheet for test Specimen A
Material of Test:
Test Load:
Scale:
Test No
1
2
3
4
5
Depth Measured
Rockwell Hardness Number
Average Rockwell Hardness Number:
Complete the following tasks:
1. Plot and discuss a bar chart (see an example in Figure 13.3) of the test results (each
test must be shown) for all the ferrous materials on a single graph.
85
2. Which of the ferrous material has the most Rockwell hardness number?
3. Plot and discuss a bar chart of the test results, each test (1-5) must be shown for all
the non-ferrous materials on a single graph.
4. Which of the non-ferrous material has the most hardness number?
5. Can you explain why the depth of the indentation does not return to h0 after the
additional load is removed in loading stage 3?
6. Can you explain why the depth measured at different point a given sample may not
the same?
7.
Briefly explain the difference between a Rockwell hardness test, a Brinell hardness
test, and a Vickers hardness test.
8. State the safety/precaution measures taken to ensure that the results are accurate
Ferrous Materials
Test 5
Test 4
Test 3
Test 2
Test 1
Material type
Sample B
Average harndess for material type B
Test 5
Test 4
Test 3
Test 2
Test 1
Sample A
0
10
Average harndess for material type A
20
30
40
50
60
70
Rockwell hardness number
Figure 13.3 A Sample of the bar chart
13.6 Conclusion
86
14 TENSILE TESTING
Equation Chapter (Next) Section 1
14.1 Introduction
The strength of a material is often of primary concern. This is the reason why a lot of
attention is given to it. The strength of interest may be measured in terms of either the
stress necessary to cause appreciable plastic deformation or the maximum stress that the
material can withstand.
The Objectives of the tensile testing experiment are to determine: (a) The ultimate tensile
strength of materials; (b) The percentage elongation; and (c) The percentage reduction in
area.
14.2 Apparatus
A universal testing machine (UTM) is typically used to carry out tensile tests. A universal
testing machine is capable of carrying out material tests in tension, compression, and
bending. UTMs are either electromechanical or hydraulic. The principal difference between
both is the method by which the load is applied.
Figure 14.1 shows the universal testing machine to be used in this course. It is made up of
the upper, middle and lower cross heads. It has an oil sump for storing the hydraulic fluid
used to lift the cross heads. This system has got a computer attached to it, for digitally
monitoring the results of the test.
Figure 14.2 shows the dimensions of a typical tensile test specimen. The gauge length is the
most important part of the specimen.
87
Upper cross-head
Screw column
Screw column
Desktop with software
installed
Middle cross head
lower cross head
Piston
Oil sump
Figure 14.1 The Universal t Test Apparatus
Figure 14.2 Test Specimen for Tension Test
14.3 Theory
When a body is subjected to tensile forces, there is an elongation in the length of the
material, usually this elongation is accompanied by a contraction in the cross-sectional area
of the material. The tensile force is recorded on the machine as a function of the increase
in gage length. Such plots of tensile force versus tensile elongation would be of little value
if they were not normalized with respect to specimen dimensions. The advantage of dealing
with stress versus strain rather than load versus elongation is that the stress-strain curve is
virtually independent of specimen dimensions.
88
The stress-strain diagram for a typical brittle and a typical ductile material is shown on
Figure 14.3. The portion 𝑂𝑎 is the proportional (elastic) limit. Point b is the upper yield
stress point while point c is the lower yield stress point. Point e gives the ultimate
strength/stress point while point f is the breaking or rupture point.
Figure 14.3 Stress-strain diagram
The Engineering stress (𝑠) is determined from Equation (14.1). 𝐹 is the tensile force
applied to the material, 𝐴0 is the initial cross-sectional area at the gauge length. This 𝐴0
value is fixed.
𝑆=
𝐹
𝐴0
(13.1)
The engineering strain (𝑒 ) is determined from Equation (14.2). ∆𝐿 is the change in length
of the specimen. 𝐿0 is the initial gauge length of the material. This 𝐿0 value is fixed.
𝑒=
∆𝐿
𝐿0
(13.2)
In real practice, the cross-sectional area used for determining the stress is not constant
after the initial extension. Likewise, the length of the specimen does not remain constant
after the initial elongation, so therefore there is a need to properly define the True stress
and strain in the material.
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The true stress is defined using Equation (14.3). 𝐴 is the instantaneous area of the
specimen when the force is applied on it.
𝜎=
𝐹
𝐴
(13.3)
The true strain is determined from Equation (14.4).
𝜀 = ln [
𝐿
]
𝐿0
(13.4)
Howbeit, the true stress can be determined from the engineering stress and strain by using
Equation (14.5).
𝜎 = 𝑆(1 + 𝑒)
(13.5)
The true strain can be determined from the engineering stress as:
𝜀 = ln [1 + 𝑒]
(13.6)
It is important to note that at very low strains, the differences between true and
engineering stress and strain are very small. It does not really matter whether Young’s
modulus is defined in terms of engineering or true stress strain.
The practical way for determining the yield strength of a material from the Engineering
stress-strain diagram is by drawing a parallel line (see Figure 14.4) to the proportional
limit, that originates with an offset strain of 0.2% (0.002). The point at which the parallel
line intersects the engineering stress versus engineering strain graph is the yield strength
of the material.
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Figure 14.4 Method for determining the yield strength of a material12
14.4 Test Procedures
Different materials have been provided for the tensile testing. The general procedure for
carrying out a Tensile Test on the UTM Machine is highlighted under two broad headings
below:
Procedure One: Preparation of the Test Sample and the UTM
1. Prepare the work piece to be tested and mark out the gauge length appropriately.
2. Boot the computer and open the UTM Software.
3. Ensure there is a connection between the computer and machine. This is shown by
the electronic indicator displaying "PC" .
4. Ensure that the Load valve is closed before opening the pressure release valve.
5. Ensure that the Cross-Head moves up and down when the machine is ON as this will
enable the engineer fix the work piece.
6. Fix the workpiece on the machine's jaws, adjust the crosshead up or down to ensure
it is clamped appropriately.
7. Switch on the Hydraulic System by pressing the ON button. Allow to warm for a while
and stop after approximately 3-5mins by pressing the OFF button.
12
Source: ASM International “ Introduction to Tensile Testing”, 2004. Second Edition. www.asminternational.org
91
8. On the UTM Software Click on File - - create New file and input the adequate
parameters such as (a) file name, (b) select the material type to be tested(eg. Round
solid).
Procedure Two: Running the test and retrieving the test results
1.
2.
3.
4.
5.
Ensure that the pressure release and load valves are closed before starting the test.
Click on TARE to reset load readings.
Click on Start Test and then ON the hydraulic button.
Slowly set the load valve to a desirable loading rate in KN.
On fracture, the machine automatically shuts off and the software automatically
opens an “after test data entry” window.
6. On the window interface, enter the measured (a) final gauge length and, (b) final
neck diameter of the work piece.
7. After the above is achieved, the test result displays showing the graph and other
results. Hence, the test results can be exported to csv format for data reading and
analysis.
14.5 Results and Discussions
The data and the computations for each material type tested can be presented on Table
14-1. A similar table should be drawn for each material type tested.
Table 14-1 Sample data sheet for tensile testing of material.
Material Type: __________________________________
Diameter of material: ________________________ (cm)
Initial cross sectional area (𝐴0 ) of the material: ______________ ((𝑚2 )
Original length (𝐿0 ) of the material: ___________________ (cm)
S/No
Force
(𝑁)
Extension
( 𝑚𝑚)
Engineering
stress
(𝑆) in
𝑁/𝑚2
Engineering
Strain
True stress
(𝜎) in
𝑁/𝑚2
True strain
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Complete the following tasks:
1. On a single graph sheet, plot and discuss a graph of load versus extension for all the
materials tested.
2. On a single graph sheet, plot and discuss a graph of engineering stress versus
engineering strain; and a graph of true stress versus true strain for the material
tested.
3. Determine the yield strength of each material.
4. Determine the young modulus of elasticity of each of the material using:
a. The engineering stress versus engineering strain plot in (2) above
b. The true stress versus true strain plot in (3) above.
5. Is there any difference between the young modulus of elasticity determined using
method 5(a) and 5(b).
6. Determine the ultimate strength of each of the material.
14.6 Conclusions
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Appendix A Difference between Flat and Vee belt
Difference between Flat and V-Belt13
13
https://www.google.com/imgres?imgurl=http%3A%2F%2Fwww.difference.minaprem.com%2Fwpcontent%2Fuploads%2F2019%2F06%2FDifferences-between-flat-belt-drive-and-V-beltdrive.jpg&imgrefurl=http%3A%2F%2Fwww.difference.minaprem.com%2Fsolid%2Fdifferencebetween-flat-belt-drive-and-v-belt-drive%2F&tbnid=B1cdsz0HqnJ8M&vet=12ahUKEwiAurTp_cDnAhUV4RoKHSbfCcsQMygFegUIARDzAQ..i&docid=uMaFfzbr4bx1M&w=1495&h=976&q=application%20of%20flat%20belts&ved=2ahUKEwiAurTp_cDnAhUV4Ro
KHSbfCcsQMygFegUIARDzAQ
94
Appendix B Data for Tensile Testing
95