WBL1: Using Engineering Science and Mathematics Report

This report presents a comprehensive exploration of engineering science, addressing key areas such as mathematics, mechanics, the strength of materials, thermodynamics, and electro-technology. The mathematics section covers arithmetic, algebra, logarithms, graphs, trigonometry, mensuration, calculus, differentiation, and integration, with solved problems illustrating each concept. The mechanics section delves into statics, dynamics, and kinematics, including detailed explanations and solved problems related to beams, supports, loads, and motion analysis. The strength of materials section discusses essential properties like stiffness, strength, stability, and durability, with an introduction to Mohr's circle and beam deflection. Thermodynamics principles, including the first law of thermodynamics, are presented. The electro-technology section is also included. The report aims to apply theoretical knowledge to practical engineering scenarios, providing a solid foundation for understanding and solving complex engineering problems. The reflective review at the end summarizes the learning outcomes and practical applications of the topics covered.

Assignment 1
Contents
A. Mathematics:..........................................................................................................................................2
B. Mechanics:..............................................................................................................................................5
C. The strength of materials:.....................................................................................................................13
D. Thermodynamics:.................................................................................................................................16
E Electro-technology.................................................................................................................................24
F. Gantry Crane Design:............................................................................................................................30
Reflective Review:....................................................................................................................................40
References:................................................................................................................................................43

A. Mathematics:
1. Arithmetic:
The topics in the Arithmetic is subdivided into:
a) Addition and subtraction
b) Multiplication and division
c) Negative numbers
d) Fractions
e) Decimals
a) Addition and subtraction:
In arithmetic addition and subtraction simple addition and subtraction of Natural
numbers, Whole numbers, Complex numbers, etc. Will be performed.
Solved problems:
Using place value to add three digit numbers:
536 500 +30 + 6
+398 300+ 90+ 8
934 900 + 30 + 4 = 934
Multi digit SUBTRACTING three digit numbers with regrouping:
971 900 +70 + 1 900 +60 +11
−659 300+ 90+ 8 600- 50- 9
312 300+10+2 = 312
In regrouping the numbers which cannot be subtracted from a smaller number must
be regrouped by taking a 10 from the number which is next to it and then the
regrouped number will be subtracted. (Simpson, S. G. 2017).)

b) Multiplication and division:
Multiplication is a mathematical operation performed on two numbers to achieve
the 3rd number, the numbers which are being operated is known as an operand and
the derived result is known as a product.
Multi digit Multiplication with Regroup:
253 X 7=1771 | 7(3) = 21 : 7 (5) = 35 : 7(2) = 14
The division is the process of splitting a certain quantity into equal numbers. Let us
say if there are 6 apples and 2 persons need to take them in equal quantity then the
answer will be 3, each can take 3 apples.
7182
42 =171
c) Negative numbers:
The subtraction of two negative numbers should be operated with addition but the
final solution should get a negative symbol.
2. Algebra:
Algebra is a division of mathematics which will solve the problem with the
inclusion of letters and other symbols which is used to represent certain quantities.
Problems such as irrational numbers, quadratics, sequences, system equations,
linear equations, etc. can be solved with algebraic methods.
3. Logarithms:
A log value is a quantity to which an entity must be raised to obtain a certain value.
Let us say that the 10 must be raised to power 2 to get 100.
4. Graphs:

Graphs are a mathematical representation of the relationship between two objects.
A graph is a graphical representation of two or more different quantities which
must be related to each other.
5. Trigonometry:
The branch of mathematics which deals relationship among the angles and sides of
a triangle and the appropriate function governing any angle.
6. Mensuration:
Mensuration is the act of making a measurement of a particular geometry such as
length, area, and volumes.
7.Calculus:
Calculus is a branch of mathematics which deals with varying quantities, calculus
is a method of calculation which implements different symbols and notations.
8. Differentiation:
Differentiation is the change of function whereas the calculus is the change of
derivative. (Jia, D., et.al 2017) (Sing, K., 2003).
d
dx ( X ) =X¿
9. Calculus:
Calculus is a branch of mathematics which deals with varying quantities, calculus
is a method of calculation which implements different symbols and notations.
(Bird, J.O., 2010).
10. Integration:
Integration is also known as anti-differentiation. It is the reversed process of
differentiation, if F’X = f(x) then, f(x) is the integration of F’X.

Basic integration formulae:
∫ X dx= X2
2 +C
B. Mechanics:
Mechanics is the stud
Application of mechanics in Marine Engineering
1. Statics:
Statics is concerned with forces that act on a body during equilibrium
condition i.e. when it is stationary. The shape of the body does not have any
influence with respect to the external concurrent forces applied on the
surface of the body hence it is considered as a particle. If the forces are non-
concurrent then the body is considered as a plane rigid body. (Anderson, T.
L. 2017).
To understand the static properties when the forces are non-concurrent, coplanar a
free body diagram representing magnitude, forces, moments etc. is to be drawn.
Steps for drawing a free body diagram. (Mike Tooley., 2012).
 Discern the object to be drawn. (If there are multiple objects of interest,
then multiple diagrams must be drawn.)
 Distinguish the forces acting directly and the object exerting that forces.
 The object of interest is denoted by dot.
 Each force which is to be drawn in the direction of the force being exerted
represented by vector
 Stationary object with constant velocity: Vectors graphically added up to 0
 Accelerating object: Sum of vector to produce equal direction as
acceleration.

Beam, Support and Load
Beam
We are quite aware of the need for beams in the field of engineering. A beam is
considered as a structural member that is capable of withstanding loads acting
perpendicular to the axis of the beam.
Support
Supports are provided at the end of the beams where the external loads are
transferred and the beam stays in a state of equilibrium.
Types of supports
 Fixed support
 Hinged support
 Roller support
Types of loads
 Point load
 UVL Uniformly varying load
 UDL Uniformly Distributed load
Problem:
A ship having a parallel post-PQRS is subjected to parallel forces. At what distance
from point P an equivalent single force acts that will give the same moment taken
about at point P? Also, reduce the system of forces to
I. A single force and a couple about the z-axis at Point P
II. A single force and a couple about the z-axis at Point S
MZ-AXIS AT POINT P = -150(2)+200(5)-25(7)
= 525 kN-m

Resultant force R= 50-150+200-25= 75Kn
Moment due to 75kN acting at x distance
75 (x) = 525
x=7 m
MZ-AXIS AT POINT Q = -50(2)+200(3)-25(5)
= 375 kN-m
MZ-AXIS AT POINT P = -150(2)+200(5)-25(7)
= 525 kN-m
MZ-AXIS AT POINT S = 150(5)+200(2)-50(7)
= 0
2. Dynamics:
A fixed and unchanged space where the shape of a rigid body is defined The
geometric centre of the rigid body lies at the origin of the body space. The
shape of the body does not have any influence with respect to the external
concurrent forces applied on the surface of the body hence it is considered as
a particle. If the forces are non-concurrent then the body is considered as a
plane rigid body

To understand the dynamic properties when the forces are non-concurrent,
coplanar a free body diagram representing magnitude, forces, moments etc. is to be
drawn.
Steps for drawing a free body diagram.
 Discern the object to be drawn. (If there are multiple objects of interest,
then multiple diagrams must be drawn.)
 Distinguish all the forces acting directly on the object and the object
exerting them.
 A dot represents the object of interest.
 A vector represents each force which is to be drawn in the direction of the
force being exerted.
 If the object is stationary or is moving at a constant velocity, the vectors
should graphically add up to zero. If the object is accelerating, the sum of
the vectors should produce a vector in the same direction as the acceleration.
Position and orientation

Problem
The anchor of a ship attached at the side rotates during movement about an axis
and the equation of motion isθ=¿t4-14t2 – 50 where θ is expressed in radians and t
in seconds. Determine position, velocity and acceleration when t=3sec.
θ=¿t4-14t2 – 50
θ = (3)4-14(9)-50= -95 rad
ω= 4t3-28t
=4(9)(3)-28(3) =24 rad/s
α= 12t2-28
=12(9)-28 = 81 rad/s2
3. Kinematics
Kinematics is concerned with the motion of a body without the inclusion of
forces which cause the motion. (Anderson, T. L. 2017).
Equations involved

Problem:
A flare is shot vertically upwards from a ship, the initial velocity of shotted
flare is 30 m/sec. Idealized motion assumed, what will be the rise, How long
will it be before it returns to the starting point and when will it reach its peak
flight and what would be the velocity be at that time Given: vo = +30
meters/sec (upward positive). At its peak the ball stops; this infers that the
speed at the peak is zero, speak = 0 when t peak=time in order to reach the peak.
Then the ball starts downward with g = -9.80 m/sec2.
Relationships: vf =v0 +¿
s=vo t+ ¿2
2
2 g • s=vf
2 – vo
2
,the original speed vo and g, w
we get 0=30.0 m/s−9.80 m/sec 2
tpeak solving for tpeak.
If velocity positive upward: g gravity acceleration is negative (downward).
time taken inorder to reach the peak,
tpeak = 30.0 meters/9.80 m/ sec2 = 3.06 seconds
Substitute

s=vo t+ ¿2
2 = 30 m/sec • 3.06 sec - 1/2 (9.8 m/ sec2 )(3.06)2
2 g • s=vf
2 – vo
2
s = +45.9 m
The FLARE will take the same time to rise as to fall;
Total flight time,
= 2(3.06 sec) = 6.12 sec.
Final velocity
vf = 30 meters/sec - 9.80 mete/ sec2• 6.12 sec;
vf = -30.0 meters/sec
4. Kinetics:
Kinetics is a branch of the Machines theory which is responsible to relate the
action of forces on bodies to their resulting motion. It is the study of motion of a
body without considering the forces acting on it.
Solved Problems:
A drop-valve of ship engine is closed with the help of a spring after the trip gear
operation. The spring stiffness N is given as that it require 4N force per mm of
compression. When opened completely the . When closed the compression is 30
mm. The mass of the valve is 5 kg and the resistance may be taken as constant and
equal to 70 N. Find the time taken to close the valve after the operation of the trip.
: s=4 N
mm =4000 N
m ; x 1=75 mm=0.075 m;
x 2=30 mm=0.03 m;
mass = 5 Kg
R : 70 Newton
Let,

X: valve Displacement
When the valve closed,
The value of x=x 1 – x 2
¿ 0.075 – 0.03 = 0.045 meters
Push of spring:
Q = 4000 (0.075 – X ) N
Downward force: P
P=Q+ m. g – R
¿ 279 – 4000 x
Also Force, P = Mass × Acceleration
On integrating and solving the equation we get
t = 1.2 /( 800 )1.2 / 28.3 =0.0424 s
C. The strength of materials:
The strength of the material is concerned with a structure which must be durable
and reliable sometimes economical in the application process. The reliability is