Physics
CONTENTS
2FOREWORD - SI UNITS
2.1 matter4
2.1.1.1The Nature of Matter4
2.1.2.1Chemical compounds5
2.1.3.1states of matter6
2.1.4.1Changes between states7
2.2 MECHANICS9
2.2.1STATICS9
2.2.1.1FORCES9
2.2.1.2CENTRE OF GRAVITY17
2.2.1.3STRESS20
2.2.1.4PROPERTIES OF MATTER26
2.2.1.5PRESSURE AND BUOYANCY IN LIQUIDS28
2.2.2KINETICS30
2.2.2.1LINEAR MOVEMENT30
2.2.2.2ROTATIONAL MOTION34
2.2.2.3PERIODIC MOTION35
2.2.2.4SIMPLE THEORY OF VIBRATION, HARMONICS AND RESONANCE36
2.2.2.5VELOCITY RATIO, MECHANICAL ADVANTAGE AND EFFICIENCY37
2.2.3DYNAMICS40
2.2.3.1MASS40
2.2.3.2MOMENTUM42
2.2.4FLUID DYNAMICS49
2.2.4.1SPECIFIC GRAVITY AND DENSITY49
2.2.4.2VISCOSITY51
2.3 THERMODYNAMICS56
2.3.1.1TEMPERATURE56
2.3.1.2HEAT57
2.3.2.1HEAT capacity57
2.3.2.2HEAT TRANSFER58
2.3.2.3VOLUMETRIC EXPANSION59
2.3.2.4THE LAWS OF THERMODYNAMICS61
GASES63
2.3.2.6ISOTHERMAL AND ADIABATIC PROCESSES65
2.3.2.7HEAT OF FUSION69
2.4 OPTICS (LIGHT)71
2.4.1.1SPEED OF LIGHT71
2.4.2.1LAWS OF REFLECTION AND REFRACTION71
2.4.3.1FIBRE OPTICS76
2.5 WAVE MOTION AND SOUND78
2.5.1.1WAVE MOTION78
2.5.2.1Sound84
FOREWORD - SI UNITS
Introduction
The study of physics is important because so much of life today
consists of applying physical principles to our needs. Most
machines we use today require a knowledge of physics to understand
their operation. Complete understanding of many of these principles
requires a much deeper knowledge than required by the JAA and the
JAR-66 syllabus for the licences.
A number of applications of physics are mentioned in this
chapter and, whenever you have learned one of these, you will need
to be aware of the many different places in aeronautics where the
application is used. Thus you will find that the laws, formulae and
calculations of physics are not just subjects for examination but
the main principle on which aircraft are flown and operated.
Physics is the study of what happens in the world involving
matter and energy.
Matter is the word used to described what things or objects are
made of. Matter can be solid, liquid or gaseous. Energy is that
which causes things to happen. As an example, electrical energy
causes an electric motor to turn, which can cause a weight to be
moved, or lifted.
As more and more 'happenings' have been studied, the subject of
physics has grown, and physical laws have become established,
usually being expressed in terms of mathematical formula, and
graphs. Physical laws are based on the basic quantities - length,
mass and time, together with temperature and electrical current.
Physical laws also involve other quantities which are derived from
the basic quantities.
What are these units? Over the years, different nations have
derived their own units (e.g. inches, pounds, minutes or
centimetres, grams and seconds), but an International System is now
generally used - the SI system.
The SI system is based on the metre (m), kilogram (kg) and
second (s) system.
base units
(
)
second)
per
second
per
(metre
s
m
s
s
m
2
=
To understand what is meant by the term derived quantities or
units consider these examples; Area is calculated by multiplying a
length by another length, so the derived unit of area is metre2
(m2). Speed is calculated by dividing distance (length) by time ,
so the derived unit is metre/second (m/s). Acceleration is change
of speed divided by time, so the derived unit is:
Some examples are given below:
Basic SI Units
Length
(L)
Metre
(m)
Mass
(m)
Kilogram
(kg)
Time
(t)
Second
(s)
Temperature;
Celsius(()
Degree Celsius(C)
Kelvin(T)
Kelvin
(K)
Electric Current(I)
Ampere
(A)
Derived SI Units
Area
(A)
Square Metre
(m2)
Volume
(V)
Cubic Metre
(m3)
Density
(()
Kg / Cubic Metre
(kg/m3)
Velocity
(V)
Metre per second
(m/s)
Acceleration(a)
Metre per second per second(m/s2)
Momentum
Kg metre per second
(kg.m/s)
derived units
Some physical quantities have derived units which become rather
complicated, and so are replaced with simple units created
specifically to represent the physical quantity. For example, force
is mass multiplied by acceleration, which is logically kg.m/s2
(kilogram metre per second per second), but this is replaced by the
Newton (N).
Examples are:
Force
(F)
Newton
(N)
Pressure
(p)
Pascal
(Pa)
Energy
(E)
Joule
(J)
Work
(W)
Joule
(J)
Power
(P)
Watt
(w)
Frequency
(f)
Hertz
(Hz)
Note also that to avoid very large or small numbers, multiples
or sub-multiples are often used. For example;
1,000,000= 106 is replaced by 'mega'(M)
1,000
= 103 is replaced by 'kilo'
(k)
1/1000= 10-3 is replaced by 'milli'
(m)
1/1000,000= 10-6 is replaced by 'micro'(()
2.1 matter
2.1.1.1 The Nature of Matter
Matter is defined as anything that occupies space. Matter is
made of tiny particles called molecules which are too small to be
seen with the naked eye, but they can be observed with an electron
microscope. When a molecule is viewed under an electron microscope
it can be seen to consist of even smaller particles called atoms
and can be seen to be in continuous motion.
Atoms are the smallest particles of matter that can take part in
a chemical reactions but they are themselves constructed of even
smaller atomic particles.
The Structure of an Atom
A hydrogen atom is very small indeed (about 10 10 m in
diameter), but if it could be magnified sufficiently it would be
seen to consist of a core or nucleus with a particle called an
electron travelling around it in an elliptical orbit.
The nucleus has a positive charge of electricity and the
electron an equal negative charge; thus the whole atom is
electrically neutral and the electrical attraction keeps the
electron circling the nucleus. Atoms of other elements have more
than one electron travelling around the nucleus, the nucleus
containing sufficient positive charges to balance the number of
electrons.
The Nucleus
The particles in the nucleus each carrying a positive charge are
called protons. In addition to the protons the nucleus usually
contains electrically neutral particles called neutrons. Neutrons
have the same mass as protons whereas electrons are very much
smaller, only EQ \f(1,1836) of the mass of a proton.
There are currently 111 known elements or atoms. Each has an
identifiable number of protons, neutrons and electrons. Every atom
has its own atomic number, as well as its own atomic mass (refer to
Fig.2). The atomic number is calculated by the element number of
protons, and the atomic mass by its number of nucleons, (protons
and neutrons combined).
1
H
1.00
Atomic no.
Symbol
Atomic mass
3
Li
6.94
4
Be
9.01
11
Na
22.9
12
Mg
24.3
19
K
39.0
20
Ca
44.0
21
Sc
44.9
22
Ti
47.8
23
V
50.9
24
Cr
52.9
25
Mn
54.9
26
Fe
55.8
27
Co
58.9
37
Rb
85.4
38
Sr
87.6
39
Y
88.9
40
Zr
91.2
41
Nb
92.9
42
Mo
95.9
43
Tc
98.0
44
Ru
101.1
45
Rh
102.9
Figure 2.An extract from the periodic table.
Neutrons
The neutron simply adds to the weight of the nucleus and hence
the atom. There is no simple rule for determining the number of
neutrons in any atom. In fact atoms of the same kind can contain
different numbers of neutrons. For example chlorine may contain 18
20 neutrons in its nucleus.
The atoms are chemically indistinguishable and are called
isotopes. The weight of an atom is due to the protons and neutrons
(the electrons are negligible in weight), thus the atomic weight is
virtually equal to the sum of the protons and the neutrons.
Electrons
The electron orbits define the size or volume occupied by the
atom. The electrons travel in orbits which are many times the
diameter of the nucleus and hence the space occupied by an atom is
virtually empty! The electrical properties of the atom are
determined by how tightly the electrons are bound by electrical
attraction to the nucleus.
2.1.2.1 Chemical compounds
When atoms bond together they form a molecule. Generally there
are two types of molecules. Those molecules that consist of a
single type of atom, for example the hydrogen normally exists as a
molecule of two atoms of hydrogen joined together and has the
chemical symbol H2. A molecule that consists of a single element is
called a monatomic molecule. All other molecules are made up of two
or more atoms and are known as chemical compounds.
When atoms bond together to form a molecule they share
electrons. Water (H2O) is made up of two atoms of hydrogen and one
atom of oxygen. In the example of H2O the oxygen atom has six
electrons in the outer or valence shell (refer to Fig. 3). As there
is room for eight electrons, one oxygen atom can combine with two
hydrogen atoms by sharing the single electron from each hydrogen
atom.
2.1.3.1 states of matter
Matter is composed of several molecules. The molecule is the
smallest unit of substance that exhibits the physical and chemical
properties of the substance. Furthermore, all molecules of a
particular substance are exactly alike and unique to that
substance.
All matter exists in one of three physical states, solid, liquid
and gas. A physical state refers to the condition of a compound and
has no affect on a compound's chemical structure. So ice water and
steam are all H2O, and the same type of matter appears in all these
states.
All atoms and molecules in matter are constantly in motion. This
motion is caused by the heat energy in the material. The degree of
motion determines the physical state of the matter.
As well as being in continuous motion, molecules also exert
strong electrical forces on each other when they are close
together. The forces are both attractive and repulsive. Attractive
forces hold matter together: repulsive forces cause matter to
resist compression. All the internal forces in matter are
summarised in the kinetic theory, which also explains the existence
of the solid, liquid and gaseous states.
Solid. A solid has definite mass, volume and shape.
The kinetic theory states that in solids the molecules are close
together and the attractive and repulsive forces between
neighbouring molecules balance: the molecules vibrate about a fixed
position.
Liquid. A liquid has definite mass and volume but takes the
shape of its container.
The molecules in a liquid are slightly farther apart than in a
solid but close enough together to have a definite volume. As well
as vibrating they are free to move over short distances in all
directions.
Gas.A gas has definite mass but takes the volume and shape of
its container.
The molecules in a gas are much farther apart in a gas than in
solids or liquids. They dash around at very high speeds in the
space available to them and it is only when they impact on the
walls of their container that the molecular forces are seen to
act.
2.1.4.1 Changes between states
In general it is possible for matter that exists in one state to
be changed into either of the other two states. But how can this be
done?
Well, ice, water and water vapour are different forms of one
type of matter, i.e. H2O molecules. The obvious difference in each
of these states is the temperature and it is this that determines
which of the three states matter will take.
Any increase in the temperature of a solid substance will
increase the energy of its molecules. The increased energy enables
the molecules to overcome each others attractive forces, until
eventually they are able to move freely as in a liquid. Further
increases in temperature give the molecules even more energy,
eventually they are able to leave the surface of the liquid in the
form of a gas.
The opposite is true if we take a gas and reduce its
temperature. In this case the reduced temperature robs the
molecules of some of their energy causing them to first slow down
and form a liquid and finally to become trapped by the attractive
forces of neighbouring molecules and forming a solid.
2.2 MECHANICS
2.2.1 STATICS
2.2.1.1 FORCES
dimension
original
dimension
in
change
L
X
If a Force is applied to a body it will cause that body to move
in the direction of the applied force, a force has both magnitude
(size) and direction. Normally more than one force acts on an
object. An object resting on a table is pulled down by its weight W
and pushed back upwards by a force R due to the table supporting
it. Since the object is stationary the forces must be in balance,
i.e. R = W, see figure 4.
Friction and air resistances are the forces that cause an object
to come to rest when the force causing the movement stops, figure
4(c). If these forces were absent, then a object, once set in
motion would continue to move with constant speed in a straight
line, figure 4(b). This is summarised by Newtons first law of
motion:
If the forces acting on an object are not in balance, i.e. there
is a net (resultant) force, they cause a change of motion, i.e. the
body accelerates or decelerates. This is known as Newtons second
law of motion:
Where
F
=
Force applied to the object
M
=
Mass of the object
a
=
Acceleration of the object
The unit of Force is the Newton. One Newton is defined as the
force which gives a mass of 1 kg an acceleration of 1 m/s2, i.e. 1
N = 1 kg m/s2.
Note. If the forces applied to an object are in balance and so
there is no change in motion there may be a change in shape. In
that case internal forces in the object (i.e. forces between
neighbouring atoms) balance the external forces. This is important
when analysing the behaviour of materials.
VECTORS AND SCALARS
Quantities are thought of as being either scalar or vector. The
term scalar means that the quantity possesses magnitude ONLY and
examples include mass, time, temperature, length etc. These
quantities may only be represented graphically to some form of
scale
Temperature Scale, 10mm = 2o
Vector quantities possess both magnitude AND direction, and if
either change the vector quantity changes. Vector quantities
include force, velocity and any quantity formed from these.
A force is a vector quantity, and as such possesses magnitude
and direction. The most convenient method is to represent the force
by means of a vector diagram as shown in figure 5.
ADDING FORCES
Two or more forces may act upon the same point so producing a
resultant force. If the forces act in the same straight line the
resultant is found by simple subtraction or addition, see figure
6.
If the forces are do not act in a straight line then they can be
added together using the parallelogram law.
If two forces acting at a point are represented in the size and
direction by the sides of a parallelogram drawn from the point,
their resultant is represented in size and direction by the
diagonal of the parallelogram drawn from the point, see figure
7.
The magnitude of the resultant force can be derived either
graphically or mathematically.
The graphical method
To use the graphical method will require a scale drawing of
forces in question, see worked example.
Worked example
Find the resultant of two forces of 4.0 N and 5.0 N acting at an
angle of 450 to each other.
Using a scale of 1.0 cm = 1.0 N, draw parallelogram ABCD with AB
= 5.0 cm, AC = 4.0 N and angle CAB = 450, see figure 8.
Figure 8
By the parallelogram law, the diagonal AD represents the
resultant in magnitude and direction; it measures 8.3 cm and angle
BAD = 210. Therefore the resultant is a force of 8.3 N acting at an
angle of 210 to the force of 5.0 N.
Triangle of Force
Considering figure 8 it can be seen that CD = AB. It is
therefore possible to find the resultant to our two forces by
drawing a triangle of forces, using the known forces as two sides
and the resultant as the third. See figure 9.
Figure 9A triangle of forces
Equilibrium
If a third force, equal in length but opposite in direction to
the resultant is added to the resultant, it will cancel the effect
of the two forces. This third force would be termed the
Equilibrium, see figure 10.
Figure 10
Polygon of forces
If three or more forces are acting on a point then the overall
resultant may be resolved by firstly applying the parallelogram law
to two of the forces, F 1 and F 2 below produce R 1. The next
force, F 3, is then resolved with the first resultant, R 1, to
produce a new resultant R 2, thus producing a polygon of forces.
This procedure can be repeated any number of times.
Mathematical solution
A single force can be seen to consist of a horizontal component
and a vertical component, which are at right angles to each
other.
If the angle between the vector of the force and the horizontal
component is ( then, trigonometry tells us that:
The vertical component = Force x sin (
The horizontal component= Force x cos (
So if there are several vectors each can be resolved into two
components.
e.g.F1 in direction (1, gives F1 sin (1, and F1 cos (1
F2 in direction (2, gives F2 sin (2, and F2 cos (2
F3 in direction (3, gives F3 sin (3, and F3 cos (3
and so on
Once all the forces have been resolved their components can then
be added together to give the sin and cos components of the
resultant.
NOTE:
For a complicated series of vectors it is possible that an
ambiguity may arise in the direction of the resultant, this can be
resolved by inspection of the sign of the sin and cos of (R.
Worked example: three forces acting on a mass.
First resolve each force into its vertical and horizontal
components.
Components of force F1 :Vertical component= 0
Horizontal component= 4 N
Components of force F2 :Vertical component= 5 X sin 530 = 4
N
Horizontal component= 5 X cos 530 = 3 N
Components of force F3 :Vertical component= 3 N
Horizontal component= 0
Components of resultant FR :Vertical component= 0 + 4 + 3 = 7
N
Horizontal component= 4 + 3 + 0 = 7 N
FR
= ( 72 + 72 = ( 49 + 49 = ( 98 = 9.9 N
(R = tan -1 7/7 = 45o
MOMENTS AND COUPLES
It has already been stated that if a force were applied to a
body, it would cause the body to move (accelerate) in the direction
of the applied force.
What if the body cannot move in a straight line, suppose the it
is free to rotate about some point. The applied force will then
cause a rotation. An example is a door. A force applied to the door
cause it to open or close, rotating about the hinge.
What is important to realise, is that the force required to move
the door is dependent on how far from the hinge the force is
applied. Similarly it is easier to loosen a nut with a long spanner
than a short one.
So the turning effect of a force is a combination of the
magnitude of the force and its distance from the point of rotation.
It is measured by multiplying the force by its perpendicular
distance of the line of action of the force from the fulcrum. The
turning effect is termed the Moment of a Force.
Moment (of a force) = Force x distance
In SI units, Newton metres = Newton x metres
In the diagram above a force of 5 N is applied at a distance of
3 m from the fulcrum, therefore:
Moment= 5 N x 3 m
= 15 N m
Moments and equilibrium
When several forces are concerned, equilibrium concerns not just
the forces, but moments as well. If equilibrium exists, then
clockwise (positive) moments are balanced by anticlockwise
(negative) moments.
(
)
second)
per
second
per
(metre
s
m
s
s
m
2
=
When two equal but opposite forces are present, whose lines of
action are not coincident, then they cause a rotation.
Together, they are termed a Couple, and the moment of a couple
is equal to the magnitude of a force F, multiplied by the distance
between them.
Where more than one force acts on a body, the total turning
effect is the algebraic sum of the moments of the forces. For
example, suppose it is necessary to calculate the resultant moment
of a pivot acting on a bell crank lever, refer to diagram
below.
AO=100 mm
OC=20 mm
BC=20 mm
Resultant Moment Calculation
The force of 10 N tends to rotate the lever clockwise, whereas
the other two forces tend to rotate the lever anti-clockwise.
Clearly, the 10 N force is in opposition to the other two and must
therefore be regarded as negative.
Total moment about O
=3 (AO cos 30() + 5 (OC) - 10 (OB sin 60()
=3 (0.100 cos 30() + 5 (0.02) - 10 (0.04 sin 60()
=0.2598 + 0.100 - 0.3464
=0.0134 N m in an anti-clockwise sense
Note that the direction as well as the magnitude of the total
moment is given, and that the unit of a moment is the product of
the unit of force, the Newton (N) and the unit of length, the metre
(m).
2.2.1.2 CENTRE OF GRAVITY
Consider a body as an accumulation of many small masses
(molecules), all subject to gravitational attraction. The total
weight, which is a force, is equal to the sum of the individual
masses, multiplied by the gravitational acceleration (g = 9.81
m/s2).
W = mg
The diagram shows that the individual forces all act in the same
direction, but have different lines of action. There must be datum
position, such that the total moment to one side, causing a
clockwise rotation, is balanced by a total moment, on the other
side, which causes an anticlockwise rotation. In other words, the
total weight can be considered to act through that datum
position.
If the body is considered in two different positions, the weight
acts through two lines of action, W1 and W2 and these interact at
point G, which is termed the Centre of Gravity (c of g).
A 2-dimensional body (one of negligible thickness) is termed a
lamina. Therefore the body has area only and no volume. The point G
is then termed a centroid. If a lamina is suspended from a point
P1, the centroid G will hang vertically below P1. If suspended from
P2, G will hang below P2. Position G, and therefore the c of g is
at the intersection as shown. Hence, the Centre of Gravity is the
point through which the Total Mass of the body may be considered to
act.
A regular lamina, such as a rectangle, has its centre of gravity
at the intersection of the diagonals.
A triangle has its centre of gravity at the intersection of the
medians, i.e. at the midpoint of each side.
If a lamina is irregular in shape but can be shown to be
composed of a several regular shapes, the centre of gravity of the
lamina can be deduced by splitting it into its regular sections,
calculating the moments of these areas about a given datum, and
then equating the sum of these moments to the moment of the
composite lamina.
Expressed as an algebraic formula,
W, X, + W2 X2 + W3 X3 = (W1 + W2 + W3) x G
Where G is the position of the centroid, with respect to the
datum.
(G =
W
W
W
X
W
X
W
X,
W,
3
2
1
3
3
2
2
+
+
+
+
This is the principle behind Weight and Balance.
For a 3-dimensional body, the centre of gravity can be
determined practically by several methods, such as by measuring and
equating moments, and thus is done when calculating Weight and
Balance of aircraft.
As already stated the centre of gravity of a solid object is the
point about which the total weight appears to act. Or, put another
way, if the object is balanced at that point, it will have no
tendency to rotate. In the case of hollow or irregular shaped
objects, it is possible for the centre of gravity to be in free
space and not within the objects at all. The most important
application of centre of gravity for aircraft mechanics is the
weight and balance of an aircraft.
If an aircraft is correctly loaded, with fuel, crew and
passengers, baggage, etc. in the correct places, the aircraft will
be in balance and easy to fly. If, for example, the baggage has
been loaded incorrectly, making the aircraft much too nose or tail
heavy, the aircraft could be difficult to fly or might even
crash.
It is important that whenever changes are made to an aircraft,
calculations MUST be made each time to ensure that the centre of
gravity is within acceptable limits set by the manufacturer of the
aircraft. These changes could be as simple as a new coat of paint,
or as complicated as the conversion from passenger to a freight
carrying role.
2.2.1.3 STRESS
When an engineer designs a component or structure he needs to
know whether it is strong enough to prevent failure due to the
loads encountered in service. He analyses the external forces and
then deduces the forces or stresses that are induced
internally.
Notice the introduction of the word stress. Obviously a
component which is twice the size in stronger and less likely to
fail due an applied load. So an important factor to consider is not
just force, but size as well. Hence stress, symbol sigma ( , is
load (force) divided by area (size).
Stress =
metre
in
force
applied
of
Area
newtons
in
force
force
External
2
or( =
metre
in
area
newtons
in
Force
2
For example if an area of 5 m2 is loaded with a force of 25 N 5
m then the area will be subjected to a stress of,
( =
metre
in
area
newtons
in
Force
2
=
m
5
N
25
2
= 5 N m-2.
Components fail due to being over-stressed, not over-loaded.
So long as the external forces acting on the ball, i.e.
atmospheric pressure, do not exceed the internal forces then the
ball will maintain its shape.
There are five different types of stress in mechanical
bodies.
1.
Tension
2.
Compression
3.
Torsion
4.
Bending
5.
Shear
Tension
Tension describes the force that tends to pull an object apart.
Flexible steel cable used in aircraft control systems is an example
of a component that is in designed to withstand tension loads.
Steel cable is easily bent and has little opposition to other types
of stress, but when subjected to a purely tensile load it performs
exceptionally well.
Compression
Compression is the resistance to an external force that tries to
push an object together. Aircraft rivets are driven with a
compressive force. When compression stress is applied to a rivet,
the rivet firstly expands until it fills the hole and then the
external part of the shank spreads to form a second head, which
holds the sheets of metal tightly together.
Torsion
A torsional stress is applied to a material when it is twisted.
Torsion is actually a combination of both tension and compression.
For example, when an object is subjected to a torsional stress,
tensile stresses operate diagonally across the object whilst
compression stresses act at right angles to the tensile stress.
An engine crankshaft is a component whose primary stress is
torsion. The pistons pushing down on the connecting rods rotate the
crankshaft against the opposition, or resistance of the propeller.
The resulting stresses attempt to twist the crankshaft.
Bending
In flight, the force of lift tries to bend an aircraft's wing
upward. When this happens the skin on the top of the wing is
subjected to a compressive force, whilst the skin below the wing is
pulled by a tension force. When the aircraft is on the ground the
force of gravity reverses the stresses. In this case the top of the
wing is subjected to tension stress whilst the lower skin
experiences compression stress.
SHEAR
The third stress that combines tension and compression is the
shear stress, which tries to slide an object apart. Shear stress
exists in a clevis bolt when it is used to connect a cable to a
stationary part of a structure. A fork fitting, such as drawn
below, is fastened onto one end of the cable, and an eye is
fastened to the structure. The fork and eye are held together by a
clevis bolt.
When the cable is pulled there is a shearing action that tries
to slide the bolt apart. This is a special form of tensile stress
inside the bolt caused by the fork pulling in one direction and the
eye pulling in the other.
STRAIN
Stress is a force inside an object caused by an external force.
If the outside force is great enough to cause the object to change
its shape or size, the object is not only under stress, but is also
strained.
If a length of elastic is pulled, it stretches. If the pull is
increases, it stretches more; if the pull is reduced, it
contracts.
Hookes law states that the amount of stretch (elongation) is
proportional to the applied force.
The degree of elongation or distortion has to be considered in
relation to the original length. the graph below shows how stress
varies with stress when a steel wire is stretched until it
breaks.
From point 0 to B the deformation of the wire is elastic.
A is the limit within the wire obeys Hookes law.
B is the elastic limit. Beyond this point deformation becomes
plastic.
C is the yield point. Beyond it very little force is needed to
produce a large extension.
D is the point where if the force were removed the wire would be
left permanently deformed.
E is the point were the wire breaks, it is say to have reached
its ultimate tensile stress.
The degree of distortion then has to be the actual distortion
divided by the original length (in other words, elongation per unit
length). This is termed Strain, symbol ( (epsilon). Note that
strain has no units, it is a ratio and is then expressed as a
percentage.
( = EQ \F(change in dimension,original dimension)
Example 1
Tensile strain
If a cable of 10 m length is loaded with a 100 kg weight so that
it is stretched to 11 m, what is the strain placed on the
cable?
( =
L
X
=
m
10
m
1
=
10
1
= 0.1 x 100 % = 10 %
Example 2
Compressive strain
A 25 cm rod is subjected to a compressive load so that its
length changes by 5 mm. How much strain is the rod under when
loaded?
5 mm is equivalent to 0.5 cm, therefore
( =
L
X
=
cm
25
cm
0.5
=
25
0.5
= 0.02 x 100 % = 2 %
Strain occurs in each of the stresses already mentioned in the
previous section. However, the strain involved with shearing and
torsional stresses is not expressed in the same manner above. Both
these stresses give rise to shearing action when one layer of
material moves relative to another in the direction of the applied
force. In shear strain this a straight motion in torsional strain
it is a rotational motion.
SHEAR
Shearing occurs when the applied load causes one 'layer' of
material to move relative to the adjacent layers etc.
When a riveted joint is loaded, it is a shear stress and shear
strain scenario. The rivet is being loaded, ultimately failing as
shown.
TORSION
As already mentioned torsional stress is a form of shear stress
resulting from a twisting action.
If a torque, or twisting action is applied to the bar shown, one
end will twist, or deflect relative to the other end.
Obviously, the twist will be proportional to the applied torque.
Torque has the same effect and therefore the same unit as a Moment,
i.e. Newton metres.
If the bar is considered as a series of adjacent discs, what has
happened is that each disc has twisted, or moved relative to its
neighbour, etc, etc. Hence, it is a shearing action.
The shear strain is equal to the angular deflection ( multiplied
by radius r divided by the overall length L,
( = EQ \F(r(,L)
2.2.1.4 PROPERTIES OF MATTER
DIFFUSION
The spreading of a substance of its accord is called diffusion
and is due to molecular action, e.g. a smell, whether pleasant or
not, travels quickly from its source to your nostrils where it is
detected.
Diffusion occurs in liquids and gases but not in solids. In
these two states the molecules are free to move, it is this
property that allows diffusion to occur.
SURFACE TENSION
A needle, though made of steel which is denser than water, will
float on a clean water surface. This suggests that the surface of a
liquid behaves as if it is covered with an elastic skin that is
trying to shrink.
This effect is called surface tension and it explains why small
liquid drops are always nearly spherical, i.e. a sphere has the
minimum surface area for a given volume. The surface tension can be
reduced if the liquid is contaminated, adding a detergent to the
water will cause our needle to sink.
In a liquid, the molecules still partially bond together. This
bonding force is known as surface tension and prevents liquids from
expanding and spreading out in all directions. Surface tension is
evident when a container is slightly over filled.
ADHESION and COHESION
The force of attraction between molecules of the same substance
is called cohesion, that between molecules of different substances
is called adhesion. For example, the adhesions of water to glass is
greater than the cohesion of water. Water spilt on glass wets it by
spreading out into a thin film. By contrast, water on wax forms
small spherical drops, this time the cohesion of water is greater
than the adhesion of water to wax. This fact is used in the
waterproofing of waxed garments.
CAPILLARITY
If a glass tube of small bore is dipped into water , the water
rises u the tube a few centimetres. The narrower the tube the
greater the rise . The adhesion between the glass and the water
exceeds the cohesion of the water molecules, the meniscus curves up
, and the surface tension causes the water to rise. The effect is
called capillary action.
MECHANICAL PROPERTIES OF MATTER
When selecting a material for a job need to know how it will
behave when a force acts upon it, i.e. what are its mechanical
properties.
Strength. A strong material requires a strong force to break it.
The strength of some materials depends on how the force is applied.
For example, concrete is strong when compressed but weak when
stretched, i.e. in tension.
Stiffness. A stiff material resists forces which try to change
its shape or size. It is not flexible.
Elasticity. An elastic material is one that recovers is original
shape and size after the force deforming it has been reformed. A
material that does not recover, but is permanently deformed is
plastic.
Ductility. Materials that can be rolled into sheets, drawn into
wires or worked into other useful shapes, without breaking are
ductile. Metals owe much of their usefulness to this property.
Brittleness. A material that is fragile and breaks easily is
brittle, e.g. glass and cast iron are brittle.
2.2.1.5 PRESSURE AND BUOYANCY IN LIQUIDS
Previous topics have introduced forces or loads, and then
considered stress, which can be thought of as intensity of load.
Stress is the term associated with solids. The equivalent term
associated with fluids is pressure,
so pressure = EQ \f(force,area) orp = EQ \f(F,A) .
Pressure can be generated in a fluid by applying a force which
tries to squeeze it, or reduce its volume. Pressure is the internal
reaction or resistance to that external force. It is important to
realise that pressure acts equally and in all directions throughout
that fluid. This can be very useful, because if a force applied at
one point creates pressure within a fluid, that pressure can be
transmitted to some other point in order to generate another force.
This is the principle behind hydraulic (fluid) systems, where a
mechanical input force drives a pump, creating pressure which then
acts within an actuator, so as to produce a mechanical output
force.
In this diagram, a force F1 is input to the fluid, creating
pressure, equal to EQ \f(F1,A1) throughout the fluid. This pressure
acts on area A2, and hence an output force F2 is generated.
If the pressure P is constant, then EQ \f(F1,A1) = EQ \f(F2,A2)
and if A2 is greater than A1, the output force F2 is greater than
F1.
A mechanical advantage has been created, just like using levers
or pulleys. This is the principle behind the hydraulic jack.
But remember, you don't get something for nothing; energy in =
energy out or work in = work out, and work = force x distance. In
other words, distance moved by F1 has to be greater than distance
moved by F2.
UNITS OF PRESSURE
Pressure is the measurement of a force exerted on a given area.
In the SI system pressure is expressed in Pascals (Pa) being
derived from force per unit area (Nm -2). Atmospheric pressure is
usually measured in milli-bars (mb) or pounds per square inch
(psi).
At sea level standard atmospheric pressure equals 1013.2
milli-bars or 14.69 psi at 15(C.
BUOYANCY
Archimedes Principle states that when an object is submerged in
a liquid, the object displaces a volume of liquid equal to its
volume and is supported by a force equal to the weight of the
liquid displaced. The force that supports the object is known as
the liquid's up-thrust.
For example, when a 100 cubic centimetre (cm 3) block weighing
1.5 kilograms (kg) is attached to a spring scale and lowered into a
full container of water, 100 cm 3 of water overflows out of the
container. The weight of 100 cm 3 of water is 100 grams (g),
therefore the up-thrust acting on the block is 100 g and the spring
scale reads 1.4 kg.
If the object immersed has a relative density that is less than
the liquid, the object displaces its own weight of the liquid and
it floats. The effect of up-thrust is not only present in liquids
but also in gases. Hot air balloons are able to rise because they
are filled with heated air that is less dense than the air it
displaced.
2.2.2 KINETICS
2.4 LINEAR MOVEMENT
In previous topic, we have seen that a force causes a body to
accelerate (assuming that it is free to move). Words such as speed,
velocity, acceleration have been introduced, which do not refer to
the force, but to the motion that ensues. Kinematics is the study
of motion.
When considering motion, it is important to define reference
points or datums (as has been done with other topics). With
kinematics, we usually consider datums involving position and time.
We then go on to consider the distance or displacement of the body
from that position, with respect to time elapsed.
It is now necessary to define precisely some of the words used
to describe motion.
Distance and time do not need defining as such, but we have seen
that they must relate to the datums. Distance and time are usually
represented by symbols (s) and (t) respectively.
Speed=rate of change of displacement or position
=
taken
time
travelled
distance
v
=
t
s
where v represents speed.
A word of caution - this assumes that the speed is unchanging
(constant). If not, the speed is an average speed.
If you run from your house to a friends house and travel a
distance of 1500m in 500 s, then your average speed is
500
1500
= 3 ms-1.
Similarly, if you travel 12 km to work and the journey takes 30
minutes, your average speed is
0.5
12
= 24 km h-1VELOCITY
Velocity is similar to speed, but not identical. The difference
is that velocity includes a directional component; hence velocity
is a vector (it has magnitude and direction - the magnitude
component being speed).
If a vehicle is moving around a circular track at a constant
speed, when it reaches point A, the vehicle is pointing in the
direction of the arrow which is a tangent to the circle. At point B
it's speed is the same, but the velocity is in the direction of the
arrow at B.
Similarly at C the velocity is shown by the arrow at C.
Note that the arrows at A and C are in almost opposite
directions, so the velocities are equal in magnitude, but almost
opposite in direction.
ACCELERATION
A vehicle that increases it's velocity is said to accelerate.
The sports saloon car may accelerate from rest to 96 km/h in 10s,
the acceleration is calculated from:
Acceleration=rate of change of velocity
= EQ \f(change of velocity,time)
a
= EQ \f(v2 - v1,t) where a represents acceleration.
(In the above, v, represents the initial velocity, v2 represents
the final velocity during time period t).
In the case of the car, v1 = 0 kmh-1 and v2 = 96 kmh-1,
therefore the rate of change of velocity = 96.
Acceleration =
10
96
= 9.6 kmh-1s-1
Note that as acceleration = rate of change of velocity, then it
must also be a vector quantity. This fact is important when we
consider circular motion, where direction is changing.
Remember:speed is a scalar, (magnitude only)
velocity is a vector (magnitude and direction).
If the final velocity v2 is less than v1, then obviously the
body has slowed. This implies that the acceleration is negative.
Other words such as deceleration or retardation may be used. It
must be emphasized that acceleration refers to a change in
velocity. If an aircraft is travelling at a constant velocity of
600 km/h it will have no acceleration.
2.4 EQUATION OF LINEAR MOTION
Various equations for motion in a straight line exist and can be
used to express the relationship between quantities.
If an object is accelerating uniformly such that:
u = the initial velocity and
v = the final velocity after a time t
a = EQ \f(change of velocity,time) or,
a =
t
u
-
v
This equation can be re-arranged to make v the subject:
V = u + at equation 1
If we now consider the distance traveled with uniform
acceleration.
If an object is moving with uniform acceleration a, for a
specified time (t), and the initial velocity is (u).
Since the average velocity = (u + v) and v = u + at.
We can substitute for v:
Average velocity = (u + u + at) = (2u + at) = u + at
The distance traveled s = average velocity x time = (u + at) x
t, so
s = ut + at2............................... equation 2
Using the s = average velocity x time and substituting time
=
t
u
-
v
, and
average velocity =
2
u
v
+
we have distance, s =
2
u
v
+
x
t
u
-
v
s =
2a
u
-
v
2
2
By cross multiplying we obtain
2as = v2 - u2
and finally:
v2 = u2 + 2as .............................equation 3
These are the three most common equations of linear motion.
2.4 Examples on linear motion.
An aircraft accelerates from rest to 200 km/h in 25 seconds.
What is it's acceleration in m s-2?
Firstly we must ensure that the units used are the same. As the
question wants the answer given in m/s2, we must convert 200 km
into metres and hours into seconds.
200 km = 200,000 m and 1 hour = 60 x 60 = 3,600 s, so
3,600
200,000
= 55.55 m s-1
using the equation a =
t
u
-
v
we have,
a =
25
0
-
55.55
= 2.22 m s-2
so the aircraft has accelerated at a rate of 2.22 m s-2
If an aircraft slows from 160 km/h to 10 km/h with a uniform
retardation of 5 m s-2, how long will take?
using v = u + at
160 = 10 +5t
5t = 160 10
t =
5
150
t = 30 secs
What distance will the aircraft travel in the example of
retardation in example 2?
We can use either
s = ut + at2
or
s =
2a
u
-
v
2
2
Using the latter,
s =
2.44
10
-
160
2
2
=
44
.
4
100
25600
-
= 5743 m
2.4 VELOCITY VECTORS
In exactly the same way as force vectors were added (either
graphically or mathematically), so velocity vectors can be added. A
good (aeronautical) example is the vector triangle used by pilots
and navigators when allowing for the effects of wind.
Here the pilot intended to fly from A to B (the vector AB
represents the speed of the aircraft through the air), but while
flying towards B the effect of the wind vector BC was to 'blow' the
aircraft off-course to C. So how is the pilot to fly to B instead
of C?
Obviously, the answer is to fly (head) towards D, so that the
wind blows the aircraft to B (see diagram).
Note that this is a vector triangle, in which we know 4 of the
components;
i.e.the wind magnitude and direction
the air speed (magnitude)
the track angle (direction)
The other two components may therefore be deduced, i.e. the
aircraft heading and the aircraft ground-speed. Note that the
difference between heading and track is termed drift. The aircraft
ground-speed, (i.e. the speed relative to the ground) is used to
compute the travelling time.
This is a particular aeronautical example. More generally, if
there are two vectors v1 and v2, then we can find relative
velocity.
Note the difference in terminology and direction of the arrows.
V2 relative to v1 means that to an observer moving at velocity V1,
the object moving at velocity V2 appears to be moving at that
relative velocity. (V1 relative to V2 is the apparent movement of
V1 relative to V2).
2.4 ROTATIONAL MOTION
2.4 CIRCULAR MOTION
Rotational motion means motion involving curved paths and
therefore change of direction. As with linear - motion, it may
analysed mathematically or graphically and both types of motion are
very similar in this respect, but employ different symbols. Again,
only cases of constant acceleration are considered here, and cases
involving linear translation and rotation are definitely
ignored!
Firstly, consider the equation representing rotation. They are
equivalent to those linear equations of motion.
Linear
Rotational
v2 = v1 + at
(2 = (1 + (t
s = (v1 + v2)t
(2 = ((1 + (2)t
s = v1t + at2
( = (1t + ( t2
Where ( = distance (angular displacement)
(1, (2=initial and final angular velocity
(
=angular acceleration
N.B. It is important to realise that the angular units here must
employ measurements in radians.
2.4 CENTRIPETAL FORCE
Consider a mass moving at a constant speed v, but following a
circular path. At one instant it is at position A and at a second
instant at B.
Note that although the speed is unchanged, the direction, and
hence the velocity, has changed. If the velocity has changed then
an acceleration must be present. If the mass has accelerated, then
a force must be present to cause that acceleration. This is
fundamental to circular motion.
The acceleration present = EQ \f(v2,r) , where v is the
(constant) speed and r is the radius of the circular path.
The force causing that acceleration is known as the Centripetal
Force. Now force is equal to mass times acceleration, therefore
centripetal force =
r
v
m
2
, or
r
mv
2
, and acts along the radius of the circular path, towards the
centre.
2.4 CENTRIFUGAL FORCE
More students are more familiar with the term Centrifugal than
the term Centripetal. What is the difference? Put simply, and
recalling Newton's 3rd Law, Centrifugal is the equal but opposite
reaction to the Centripetal force.
This can be shown by a diagram, with a person holding a string
tied to a mass which is rotating around the person.
Tensile force in string acts inwards to provide centripetal
force acting on mass.
Tensile force at the other end of the string acts outwards
exerting centrifugal reaction on person.
(Note again - cases involving changing speeds as well as
direction are beyond the scope of this course)
2.4 PERIODIC MOTION
Some masses move from one point to another, some move round and
round. These motions have been described as translational or
rotational.
Some masses move from one point to another, then back to the
original point, and continue to do this repetitively. The time
during which the mass moved away from, and then returned to its
original position is known as the time period and the motion is
known as periodic.
Many mechanisms or components behave in this manner - a good
example is a pendulum.
2.4 PENDULUM
If a pendulum is displaced from its stationary position and
released, it will swing back towards that position. On reaching it
however, it will not stop, because its inertia carries it on to an
equal but opposite displacement. It then returns towards the
stationary position, but carries on swinging etc, etc. Note that
the time period can be measured from a any position, through to the
next time that position is reached, with the motion in the original
direction
2.4 SPRING MASS SYSTEMS
If the mass is displaced from its original position and
released, the force in the spring will act on the mass so as to
return it to that position. It behaves like the pendulum, in that
it will continue to move up and down.
The resulting motion, up and down, can be plotted against time
and will result in a typical graph, which is sinusoidal.
2.4 SIMPLE THEORY OF VIBRATION, HARMONICS AND RESONANCE
Analysis of oscillating systems such as the pendulum or the
spring-mass will show that they often obey simple but strict laws.
For example, the instantaneous acceleration is given by the term
-(2x.
a = -(2x
(This basically states that the acceleration is proportional to
the displacement from the neutral (undisturbed) position, and in
the opposite sense to the direction of the velocity)
The constant ( is the frequency of the oscillation.
The period of the oscillation = EQ \F(2(,() .
Such motion is often referred to as Harmonic motion and analysis
reveals the pattern of such motion is sinusoidal (beyond the scope
of this course).
2.4 VIBRATION THEORY
Vibration Theory is based on the detailed analysis of vibrations
and is essentially mathematical, relying heavily on trigonometry
and calculus, involving sinusoidal functions and differential
equations.
The simple pendulum or spring-mass would according to basic
theory, continue to vibrate at constant frequency and amplitude,
once the vibration had been started. In fact, the vibrations die
away, due to other forces associated with motion, such as friction,
air resistance etc. This is termed a Damped vibration.
If a disturbing force is re-applied periodically the vibrations
can be maintained indefinitely. The frequency (and to a lesser
extent, the magnitude) of this disturbing force now becomes
critical.
Depending on the frequency, the amplitude of vibration may decay
rapidly (a damping effect) but may grow significantly.
This large increase in amplitude usually occurs when the
frequency of the disturbing force coincides with the natural
frequency of the vibration of the system (or some harmonic). This
phenomenon is known as Resonance. Designers carry out tests to
determine these frequencies, so that they can be avoided or
eliminated, as they can be very damaging.
2.4 VELOCITY RATIO, MECHANICAL ADVANTAGE AND EFFICIENCY
A machine is any device which enables a force (the effort)
acting at one point to overcome another force (the load) acting at
some other point. A lever is a simple machine , as are pulleys,
gears, screws, etc.
In the diagram below a lever lifts a load of 100 N through 0.50
m when an effort is applied at the other end. The effort can be
taken from the principle of moments about the pivot O as effort
just begins to raise the load.
clockwise moment = anticlockwise moment
effort x 2 m = load x 1m = 100 N x 1 m
effort x = 100 N m/2 m = 50 N
the lever has enabled an effort (E) to raise a load (L) twice as
large, i.e. it is a force multiplier, but E has had to move twice
as far as L. the lever has a mechanical advantage (MA) of 2 and a
velocity ratio (VR) of 2 where
MA =
E
L
andVR =
E
by
moved
distance
L
by
moved
distance
Machines make work easier and transfer energy from one place to
another. No machine is perfect and in practice more work is done by
the effort on the machine than is done by the machine on the load.
Work measure energy transfer and so we can also say that the energy
input into a machine is greater than its energy output. Some energy
is always wasted to overcome friction and some parts of the machine
itself.
efficiency =
input
energy
output
energy
=
effort
by
done
work
load
on
done
work
=
VR
MA
X 100%
this is expressed as a percentage and is always less than
100%.
2.4 LEVERS
A lever is a device used to gain a mechanical advantage. In its
most basic form, the lever is a seesaw that has a weight at each
end. The weight on one end of the seesaw tends to rotate it
anti-clockwise, whilst the weight on the other end tends to rotate
it clockwise.
Each weight produces a moment or turning force. The moment of an
object is calculated by multiplying the object's weight by the
distance the object is from the balance point or fulcrum.
A lever is in balance when the algebraic sum of the moments is
zero. In other words, a 10 kilogram weight located 2 metres to the
left of the fulcrum has a negative moment (anti-clockwise), 20
kilogram metres. A 10 kilogram weight located 2 metres to the right
of the fulcrum has a positive moment (clockwise), of 20 kilogram
metres. Since the sum of the moments is zero, the lever is
balanced.
2.4 First Class Lever
This lever has the fulcrum between the load and the effort. An
example might be using a long armed lever to lift a heavy crate
with the fulcrum very close to the crate, the effort E is applied a
distance L from the fulcrum .
The load (resistance) R, acts at a distance I from the fulcrum.
The calculation is carried out using the formula, E x L=I x R
Although less effort is required to lift the load, the lever
does not reduce the amount of work done. Work is the result of
force and distance, and if the two items from both sides are
multiplied together, they are always equal.
2.4 Second Class Lever
Unlike the first-class lever, the second-class lever has the
fulcrum at one end of the lever and effort is applied to the
opposite end. The resistance or weight, is typically placed near
the fulcrum between the two ends.
A typical example of this lever arrangement is the wheel-barrow,
refer to diagram below. Calculations are carried out using the same
formula as for the first class-class lever although, in this case,
the load and the effort move in the same direction.
Third Class Lever
In aviation, the third-class lever is primarily used to move the
load a greater distance than the effort applied. This is
accomplished by applying the effort between the fulcrum and the
resistance. The disadvantage of doing this, is that a much greater
effort is required to produce movement. A example of a third-class
lever is a landing gear retraction mechanism (refer to diagram
below) where the effort is applied close to the fulcrum, whilst the
load, (the wheel/brake assembly) is at the opposite end of the
lever.
2.2.3 DYNAMICS
2.2.3.1 MASS
Contrary to popular belief, the weight and mass of a body are
not the same. Weight is the force with which gravity attracts a
body. However, it is more important to note that the force of
gravity varies with the distance between a body and the centre of
the earth. So, the farther away an object is from the centre of the
earth, the less it weighs. The mass of an object is described as
the amount of matter in an object and is constant regardless of its
location. The extreme case of this is an object in deep space,
which still has mass but no weight.
Another definition sometimes used to describe mass is the
measurement of an object's resistance to change its state of rest,
or motion. This is seen by comparing the force needed to move a
large jet, as compared with a light aircraft. Because the jet has a
greater resistance to change, it has greater mass. The mass of an
object may be found by dividing the weight of an object by the
acceleration of gravity which is 9.81 m/s2
Mass is usually measured in kilograms (kg) or, possibly, grams
(gm) for small quantities and tonnes for larger, The Imperial
system of pounds (Ibs.) can still be found in use in aviation, for
calculation of fuel quantities, for example.
FORCE
Force has been described earlier, force is the vector quantity
representing one or more other forces, which act on a body. In this
section we will see the effect of forces when they produce, or tend
to produce, movement or a change in direction.
INERTIA
Inertia is the resistance to movement, mentioned earlier when
discussing the mass of objects. This means that if an object is
stationary it remains so, and if it is moving in one direction, it
will not deviate from that course. A force will be needed to change
either of these states; the size of the force required is a measure
of the inertia and the mass of the object.
WORK
Work is done when a force moves. Consider the case where a man
applied a force to move a small car. The initial force that he
applies overcomes the cars inertia and it moves. The work that the
man has done is equal to:
Work done = force x distance moved in the direction of the
force.
If the man continues to push the car a farther distance then the
distance moved will increase and so he will have done more
work.
The unit of work is the Newton metre (Nm) or the joule,
where
1 joule = the work done when a force of 1 Newton is applied
through a distance of 1 metre
POWER
Recalling the man pushing the car, it was stated that the
greater the distance the car was pushed, the greater the work done
(or the greater the energy expended).
But yet again, another factor arises for our consideration. The
man will only be capable of pushing it through a certain distance
within a certain time. A more powerful man will achieve the same
distance in less time. So, the word Power is introduced, which
includes time in relation to doing work.
Power =
taken
time
done
work
EQ \B(= Force x \F(distance,time) = Force x speed)
The S.I. unit of power is the Watt (W), and it is the rate of
work done when 1 joule is achieved in one second
(N.B. One horsepower is the equivalent of 746 Watts)
BRAKE HORSE POWER
Engines are often rated as being of a certain brake horsepower.
This refers to the method by which their horsepower is measured.
The engine is made to do work on a device known as a dynamometer or
'brake'. This loads the engine output, whilst a reading of the work
being done can be observed from the machine's instrumentation.
SHAFT HORSE POWER
This is a similar measurement to brake horsepower, except that
the measurement is usually taken at the output shaft of a
turbo-propeller engine. The power being produced at the shaft is
what will be delivered to the propeller, when it is installed to
the engine.
ENERGY
Now clearly the man pushing the car will become progressively
more tired the further he pushes the car, the more work he does the
more energy he expends.
Energy can be thought of as stored work. Alternatively, work is
done when Energy is expended. The unit of Energy is the same as for
Work, i.e. the Joule.
Energy can exist or be stored in a number of different forms,
and it is the change of form that is normally found in many
engineering devices.
Energy can be considered in the following forms, electrical,
chemical, heat, pressure, potential, kinetic - and there are
others. The units for all forms of energy is the Joule.
Energy due to the mechanical condition or the position of a body
is called potential energy.
The potential energy of a raised body is easily calculated. If
it falls, the force acting will be its weight and the distance
acted through; its previous height. Hence, the work done equals the
weight times the height. This is also the potential energy
held.
PE = mg x h (Joules) NB: Weight equals mass times gravity.
Another form of energy is that due to the movement of particles
of some kind. This can be the water flowing in a river, driving a
mill or turbine. The moving air driving a wind turbine which is
producing electricity; or hot gasses in a jet engine, driving the
turbine, are both forms of energy due to motion, which is known as
kinetic energy.
Kinetic energy is energy of motion. The kinetic energy of an
object is the energy it possesses because of its motion. The
kinetic energy of a point mass m is given by:
KE = mv2 Note: m is in kg and v is in ms-1
The kinetic energy of an object arises from the work done on it.
This can been seen from the example of using a constant net force
to accelerate a mass from rest to a final velocity.
Work done on mass = Fd = mad = m x
t
v
f
x
2
v
f
x t = mv2 = kinetic energy
CONSERVATION OF ENERGY
One important principle underlies the conversion of one form to
another. It is known as the Conservation of Energy, which is:
Energy cannot be created or destroyed, but can be changed from
one form to another
This allows scientific equations to be derived, after
investigation and analysis involving physical experiments.
This also suggests something most of us suspect there is no such
thing as a free lunch. Put another way, you dont get anything for
nothing, and very often, you get less out than you put in. (So
somewhere losses have occurred, this is to be expected). So a
comparison between work out and work in is obviously a measure of
the systems efficiency.
Efficiency = EQ \F(Work output,Work input)
It is usually expressed as a percentage, and so will clearly
always be less than 100%..
HEAT
Heat is defined as the energy in transit between two bodies
because of a difference in temperature. If two bodies, at different
temperatures, are bought into contact, their temperatures become
equal. Heat causes molecular movement, which is a form of kinetic
energy and, the higher the temperature, the greater the kinetic
energy of its molecules.
Thus when two bodies come into contact, the kinetic energy of
the molecules of the hotter body tends to decrease and that of the
molecules of the cooler body, to increase until both are at the
same temperature.
2.2.3.2 MOMENTUM
Momentum is a word in everyday use, but its precise meaning is
less well-known. We say that a large rugby forward, crashing
through several tackles to score a try, used his momentum. This
seems to suggest a combination of size (mass) and speed were the
contributing factors.
In fact, momentum = mass x velocity.
IMPULSE OF A FORCE
Newton's Second Law shows that the effect of a force on a body
is to bring about a change in momentum in a given time. This
provides a useful method of measuring a force, but such a
measurement becomes difficult if the time taken for the change is
very small. This would be the case if a body was subjected to a
sudden blow, shock load or impact. In such cases, it may well be
possible to measure the change in momentum with reasonable
accuracy.
The time duration of the impact force may be in doubt and, in
the absence of special equipment, may have to be estimated. Forces
of this type, having a short time duration, are called impulsive
forces and their effect on the body to which they are applied, that
is the change of momentum produced, is called the impulse.
If the impact duration is very small, the impulsive force is
very large for any given impulse or change in momentum. This can be
shown by substitution into equations.
CONSERVATION OF MOMENTUM
The principle of the Conservation of Momentum states:
When two or more masses act on each other, the total momentum of
the masses remains constant, provided no external forces, such as
friction, act.
Study of force and change in momentum lead to Newton defining
his Laws of Motion, which are fundamental to mechanical
science.
The First law states a mass remains at rest, or continues to
move at constant velocity, unless acted on by an external
force.
The Second law states that the rate of change of momentum is
proportional to the applied force.
The Third law states if mass A exerts a force on mass B, then B
exerts an equal but opposite force on A.
CHANGES IN MOMENTUM
What causes momentum to change? If the initial and final
velocities of a mass are u and v,
then change of momentum = mv - mu
= m (v - u).
Does the change of momentum happen slowly or quickly?
The rate of change of momentum = m EQ \f((v - u),t)
Inspection of this shows that force F (m.a) = m EQ \f((v - u),t)
, so, a force causes a change in momentum.
The rate of change of momentum is proportional to the magnitude
of the force causing it.
Suppose a mass A overtakes a mass B, as shown below in
illustration (a). On impact, (b), the mass B will be accelerated by
an impulsive force delivered by A, whilst the mass A will be
decelerated by an impulsive force delivered by B.
In accordance with Newton's Third Law, these impulsive forces, F
, will be equal and opposite and must, of course, act for the same
small period of time. After the impact, A and B will have some new
velocities vA, and vB. By calculation, it can be proven that the
momentum before the impact equals the momentum after the
impact.
MOMENT OF INERTIA
Moment of Inertia considers the effect of mass on bodies whose
moment is rotational. This is important to engineers, because
although vehicle move from on place to another (i.e. the moment of
the vehicle is translational) many of its components are rotating
within it.
Consider two cylinders, of equal mass, but different dimensions,
capable of being rotated.
It will be easier (require less torque) to cause the LH cylinder
to rotate. This is because the RH cylinder appears to have greater
inertia, even through the masses are the same.
So the moment of inertia (() is a function of mass and radius.
Although more detailed study of the exact relationship is beyond
the scope of this course, it can be said that the M of I is
proportional to the square of the radius.
GYROSCOPES
This topic covers gyroscopes and the allied subject of the
balancing of rotating masses. Both of these topics have direct
application to aircraft operations.
Gyroscopes are used in several of an aircrafts instruments,
which are vital to the safety of the aircraft in bad weather. There
are many different components that will not operate correctly if
they are not perfectly balanced. For example wheels, engines,
propellers, electric motors and many other components must run with
perfect smoothness.
The gyroscope is a rotor having freedom of motion in one or more
planes at right angles to the plane of rotation. With the rotor
spinning, the gyroscope will possess two fundamental
properties:
1. Gyroscopic rigidity or inertia
2. Gyroscopic precession
A gyroscope has freedom of movement about axes BB and CC, which
are at 90( to the axis of rotation AA .
Rigidity
This maintains the axis of rotation constant in space. So if a
gyroscope is spinning in free space and is not acted upon by any
outside influence or force, it will remain fixed in one position.
This facility is used in instruments such as the artificial
horizon, which shows the location of the actual horizon outside,
even when the aircraft is in poor visibility.
The mounting frame can be rotated about axes AA and BB. The
gyroscope will remain fixed in space in the position it was set,
and this is known as rigidity. If the frame is rotated about axis
CC, the gyroscope will rotate until the axis of gyroscopic rotation
is in line with the axis of the frame rotation and is known as
precession.
Precession
This term describes the angular change of direction of the plane
of rotation of a gyroscope, as a result of an external force. The
rate of this change can be used to give indications such as the
turning rate of an aircraft.
The diagram below shows a gyroscope, which has now been rotated
about axis BB. It can be seen that the axis of rotation AA is now
vertical and, in line with axis CC, which is the principle of
precession. Gyroscopes will precess to allow the plane of rotation
of the rotor to coincide with the base.
To determine the direction a gyroscope will precess, follow
these guidelines.
1. Apply a force so that it acts on the rim of the rotor at
90(.
2. Move this force around the rim of the rotor so that it moves
through 90( and in the same direction as the rotor spins.
3. Precession will move the rotor in the direction that will
result in the axes of applied force and of rotation coinciding.
4. For a constant gyroscopic speed, the rate of precession is
proportional to the applied force. The opposite also applies, so
for a given force the rate of precession is inversely proportional
to rotor speed.
Determining Precession Direction
Balancing of Rotating Masses
Perhaps the most common of all the systems encountered in
mechanical engineering practice is the rotating shaft system. If
the centroid of any mass mounted on a rotating shaft, is offset
from the axis of rotation, then the mass will exert a centrifugal
force on the shaft. This force is directly proportional to the
square of the speed of rotation of the shaft, so that, even if the
eccentricity is small, the force may be considerable at high
speeds. Such a force will tend to make the shaft bend, producing
large stresses in the shaft and causing damage to the bearings as
it does so.
A further undesirable effect would be the inducement of
sustained vibrations in the system, its supports and the
surroundings. This situation would be intolerable in an aircraft,
so that some attempt must be made to eliminate the effect of the
unwanted centrifugal force.
The eccentricity of the rotating masses cannot be removed, as
they are either a result of the design of the mechanism, such as a
crankshaft, or are due to unavoidable manufacturing imperfections.
The problem is solved, or at least minimised, by the addition of
balance weights, whose out of balance centrifugal force is exactly
equal and opposite to the original out of balance force. A common
example of this is the weights put on motor car wheels to balance
them, which makes the car much easier to drive at high speed.
FRICTION
Friction is that phenomenon in nature that always seems to be
present and acts so as to retard things that move, relative to
things that are either stationary or moving slowly. How large that
frictional force is depends on the nature of the two surfaces of
the object concerned. Rough surfaces generally produce more
friction than smooth surfaces, and some materials are naturally
'slippery'. Friction can operate in any direction, but always acts
in the sense opposing motion.
The diagram shows a body (mass m) on an inclined plane. As the
angle of the plane (() is increased, the body remains stationary,
until at some particular value of (, it begins to move down the
plane. This is because the frictional force (F) opposing motion has
reached its maximum value.
FRICTION CALCULATION
At this maximum value, the force opposing motion
Fmax = mg sin (,
and the normal reaction between the body and the plane
R = mg cos (.
EQ \f(F,R) = EQ \f(mg sin (,mg cos () = tan (
This ratio EQ \f(F,R) (tan () is termed the Coefficient of
Friction. It is generally considered in mechanics to have a value
less than 1, but some materials have a 'stickiness' associated with
them which exceeds this value.
Note also that cases occur where static friction (friction
associated with stationary objects) is greater than running
friction (where objects are now in motion).
A useful example is in flying-control systems, where engineers
have to perform both static and running friction checks.
2.2.4 FLUID DYNAMICS
Fluid is a term that includes both gases and liquids; they are
both able to flow. We will generally consider gases to be
compressible and liquids to be incompressible.
2.4 SPECIFIC GRAVITY AND DENSITY
The density of a substance is its mass per unit volume. The
density of solids and liquids varies with temperature, and the
density of a gas varies with both temperature and pressure. The
symbol for density is the Greek symbol Rho (() To find the density
of a substance, divide its mass by its volume, which will give you
the mass per unit volume, or density.
Density (()=
volume
mass
For example, the liquid that fills a certain container has a
mass of 756 kilograms. The container is 1.6 metres long, 1.0 metre
wide and 0.75 of a metre deep and we want to find the liquids
density. The volume of the container is 1.6 x 1.0 x 0.75 = 1.2 m3
and the density is
(
=
1.2
756
= 630 kg m -3
As the density of solids and liquids vary with temperature, a
standard temperature of 4(C is used when measuring the density of
each. Although temperature changes do not change the mass of a
substance, they do change the volume through thermal expansion and
contraction. This volume change means that there is a change in the
density of the substance.
When measuring the density of a gas, temperature and pressure
must be considered. Standard conditions for the measurement of gas
density is established at 0(C and a pressure of 1013.25 milli-bars
(Standard atmospheric pressure).
RELATIVE DENSITY (formerly specific gravity)
It is often necessary to compare the density of one substance
with that of another. For this reason, a standard is needed from
which all other materials can be compared. The standard when
comparing the densities of all liquids and solids is water at 4(C,
and the standard for gases is air.
Relative density is calculated by comparing the weight of a
definite volume of substance with an equal volume of water. The
following formula can be used to find the relative density, of
liquids and solids.
Relative Density =
water
of
volume
equal
of
mass
substance
a
of
volume
any
of
mass
The same formulas are used to find the density of gases by
substituting air for water. As relative density is a ratio it has
no units. For example, if a certain hydraulic fluid has a relative
density of 0.8, then 1 litre of the liquid weighs 0.8 times as much
as 1 litre of water table of typical relative densities. Remember
that the relative density of both water and air is 1.
Typical Relative Densities
Solid
Liquid
Gases
Ice
0.917
Petrol
0.72
Hydrogen
0.0695
Aluminium
2.7
Jet Fuel (JP-4)
0.785
Helium
0.138
Titanium
4.4
Alcohol
0.789
Acetylene
0.898
Iron
7.9
Kerosene
0.82
Nitrogen
0.967
Copper
8.9
Synthetic Oil
0.928
Air
1.000
Lead
11.4
Water
1.000
Oxygen
1.105
Gold
19.3
Mercury
13.6
Carbon Dioxide
1.528
Table of Typical Relative Densities
Hydrometer
A device called a hydrometer is used to measure the relative
density of liquids. This device has a glass float contained within
a cylindrical glass body. The float has a weight in the bottom and
a graduated scale at the top. When liquid is drawn into the body,
the float displays the relative density on the graduated scale.
Immersion in pure water would give a reading of 1.000, so
liquids with relative density of less or more than water would
float lower or higher than it would in water.
An area in aviation where this topic is of special interest is
the electrolyte of batteries, where the relative density is an
indication of battery condition. Another is aircraft fuel, as some
aircraft are re-fuelled by weight, whilst others are re-fuelled by
volume. Knowledge of the relative density of the fuel is essential
in this case.
2.4 VISCOSITY
Liquids such as water flow very easily whilst others, such as
treacle, flow much slower under the same conditions. Liquids of the
type that flow readily are said to be mobile, and those of the
treacle type are called viscous. Viscosity is due to friction in
the interior of the liquid.
Just as there is friction opposing movement between two solid
surfaces when one slides over another, so there is friction between
two liquid surfaces even when they consist of the same liquid. This
internal friction opposes the motion of one layer over another and,
therefore, when it is great, it makes the flow of the liquid very
slow.
Even mobile liquids possess a certain amount of viscosity. This
can be shown by stirring a container of liquid, with a piece of
wire. If you continue to stir, the contents of the container will
eventually be spinning. This proves that the viscosity of the
layers immediately next to the wire have dragged other layers
around, until all the liquid rotates.
The viscosity of a liquid rapidly decreases as its temperature
rises. Treacle will run off a warmed spoon much more readily than
it will from a cold one. Similarly when tar (which is very viscous)
is to be used for roadway repairs, it is first heated so that it
will flow readily.
Some liquids have such high viscosity that they almost have the
same properties as solids. Pitch, which is also used in road
building, is a solid black substance. If we leave a block of the
material in one position, it will, eventually begin to spread. This
shows it to be a liquid with a very high viscosity.
An even more extreme case is glass. A sheet of glass stood up on
end on a hard surface, will eventually be found to be slightly
thicker at the bottom of the sheet than at the top. So although we
could call glass a liquid with an exceedingly high viscosity, we
normally consider it a solid.
The viscosity of different liquids can be compared in different
ways. If we allow a fixed quantity to run out of a container
through a known orifice, we can time it and then compare this
against another liquid, we can say which has the lower (or higher)
viscosity. Other more complex apparatus, is required to measure
viscosity more accurately.
The knowledge of the viscosity of a liquid, such as oil is
vital. Aircraft components such as engines and gearboxes depend on
lubrication to enable them to operate efficiently.
2.4 FLUID RESISTANCE
The resistance to fluid flows can be divided into two general
groups. Skin friction is the resistance present on a thin, flat
plate, which is edgewise on to a fluid flow. The fluid is slowed up
near the surface owing to the roughness of the surface and it can
be shown that the fluid is actually stationary at the surface.
The surface roughness has an effect on the streamlines that are
away from the surface and if the surface can be made smoother, the
overall friction or drag can be reduced.
The second form of resistance is known as eddies or turbulent
airflow. This can be demonstrated by placing the flat plate at
right angles to the flow. This causes a great deal turbulence
behind the plate and a very high resistance, which is almost
entirely due to the formation of these eddies.
2.4 THE EFFECTS OF STREAMLINING
When a fluid, liquid or gas is flowing steadily over a smooth
surface, narrow layers of it follow smooth paths that are known as
streamlines. This smooth flow is also known as laminar flow. If
this stream meets large irregularities, the streamlines are broken
up and the flow becomes irregular or turbulent, as may be seen when
a stream comes upon rocks on a river bed.
The introduction of smoke into the airflow in a wind tunnels or
coloured jets into water tank experiments, makes it is possible to
see these streamlines and eddies.
When a fluid flows slowly along pipe, the flow is said to be
steady and lines, called streamlines, are drawn to represent it as
in part a of the diagram below.
If the flow is very fast and exceeds a certain critical speed,
the flow becomes turbulent and the fluid is churned up. The
streamlines are no longer straight and parallel, and eddies are
formed as in part b of the diagram. The resistance to flow
increases as a results.
The behaviour of a fluid when an object is moving in it is
similar to what occurs when a fluid flows through a pipe.
If the object, e.g. a small sphere , moves slowly, then
streamlines similar to those in part a of the diagram below, will
show the apparent motion of the fluid around the object. It will be
a steady flow. If the speed of the sphere increases, a critical
speed is reached when the flow breaks up and eddies are formed
behind the sphere as in part b, the flow becomes turbulent and the
viscous drag on the sphere increases sharply.
The critical speed can be raised by changing the shape of the
object, so reducing drag and causing steady flow to replace
turbulent flow. This is called streamlining the object and part c
shows how this done for a sphere. Streamlining is especially
important in the designing of high speed aircraft and other fast
moving vehicles.
2.4 THE COMPRESSIBILITY OF FLUIDS
All fluids are compressible, so that their density will change
with pressure, but, under steady flow conditions and provided that
the changes of density are small it is often possible to simplify
the analysis of a problem by assuming the that fluid is
incompressible and of constant density. Since liquids are
relatively difficult to compress, it is usual to treat them as if
they wee incompressible for all cases of steady flow.
Gases are easily compressed and, except when changes of pressure
and, therefore, density are very small, the effects of
compressibility and changes of internal energy must be taken into
account.
2.4 STATIC AND DYNAMIC PRESSURE
2.4 Static and Dynamic pressure.
In this diagram, the pressure acting on x x1 is due to the
weight of the fluid (in this case a liquid) acting downwards.
This weight W=mg (g = gravitational constant)
But mass
=volume ( density
=height ( cross-sectional area ( density
=h.A.(
Therefore downward force =h.(.g. A. acting on A
Therefore, the pressure= EQ \f(h(g.\O(A,/), \O(A,/))
=hpg
This is the static pressure acting at depth h within a
stationary fluid of density p.
This is straightforward enough to understand as the simple
diagram demonstrates, we can "see" the liquid.
But the same principle applies to gases also, and we know that
at altitude, the reduced density is accompanied by reduced static
pressure.
We are not aware of the static pressure within the atmosphere
which acts on our bodies, the density is low (almost 1000 times
less than water). Divers, however, quickly become aware of
increasing water pressure as they descend.
But we do become aware of greater air pressures whenever moving
air is involved, as on a windy day for example. The pressure
associated with moving air is termed dynamic pressure.
In aeronautics, moving air is essential to flight, and so
dynamic pressure is frequently referred to.
Dynamic pressure= (v2 where ( = density, v = velocity.
Note how the pressure is proportional to the square of the air
velocity.
2.4 BERNOULLIS THEOREM
The Swiss mathematician and physicist Daniel Bernoulli developed
a principle that explains the relationship between potential and
kinetic energy in a fluid. All matter contains potential energy
and/or kinetic energy. In a fluid the potential energy is that
caused by the pressure of the fluid, while the kinetic energy is
that caused by the fluids movement. Although you cannot create or
destroy energy, it is possible to exchange potential energy for
kinetic energy or vice versa.
As a fluid enters a venturi tube, it is travelling at a known
velocity and pressure. When the fluid enters the restriction it
must speed up, or increase its kinetic energy. However, when the
kinetic energy increases, the potential energy decreases and
therefore the pressure decreases. Then as the fluid continues
through the tube, both velocity and pressure return to their
original values.
Bernoullis principle can be found in a carburettor and paint
spray gun. Air passing through a venturi creates a rapid drop in
pressure, which enables the atmospheric pressure to force the fluid
into the venturi, and out of the tube in the form of a fine spray
and the theory of flight.
2.3 THERMODYNAMICS
2.3.1.1 TEMPERATURE
Heat is a form of energy that causes molecular agitation within
a material. The amount of agitation is measured in terms of
temperature, which is a measure of the kinetic energy of
molecules.
In establishing a temperature scale, two conditions are chosen
as a reference. These are the points at which pure water freezes
and boils. In the Centigrade system the scale is divided into 100
graduated increments known as degrees (() with the freezing point
of water represented by 0(C and the boiling point 100(C. The
Centigrade scale was named the Celsius scale after the Swedish
astronomer Anders Celsius who first described the centigrade scale
in 1742.
In 1802 the French chemist and physicist Joseph Louis Gay
Loussac found that when you increased the temperature of a gas by
one degree Celsius, it expands by 1/273 of its original volume. He
reasoned that if a gas was cooled, its volume would decrease by the
same amount. So if the temperature was decreased to 273 degrees
below zero, the volume of the gas would also decrease to zero, and
there would be no more molecular activity. This point is referred
to absolute zero.
On the Celsius scale absolute zero is - 273(C. On the Fahrenheit
scale it is
460(F.
In the Fahrenheit system, water freezes at 32(F and boils at
212(F. The difference between these two points is divided into 180
increments. Conversion between temperature scales. An engineering
student should be able to convert from one temperature to
another:
e.g.convert F to C-Subtract 32, then multiply by EQ \f(5,9)
convert C to F-Multiply by EQ \f(9,5) , then add