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TRANSCRIPT
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