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Basic SI Units

Length (L) Metre (m)Mass (m) Kilogram(kg)Time (t) Second (s)

Derived SI Units

Area (A) Square Metre (m2)Volume (V) Cubic Metre (m3)

Density (ρ) Kg / Cubic Metre (kg/m3)Velocity (V) Metre per second (m/s)

Force (F) Newton (N)Pressure(p) Pascal (Pa)Energy (E) Joule (J)Work (W) Joule (J)Power (P) Watt (w)Frequency(f) Hertz (Hz)

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• Matter - anything that occupies space.

- consists of atoms and molecules.

• Atom - smallest particle in an element that has the properties of the


• Molecules - the combination of two or more atoms.

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• Electron - orbits define the size or volume occupied by the atom

- negatively charged.

- total number of negatively charged electrons matches

number of positively charged protons.

• Neutron - simply adds to the weight of the nucleus.

- has no charge.

• Proton - positively charged.

- has mass same as the neutrons.

• 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.


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Particles RelativeMass

Electrical Charge


Neutron 1 0 ( zero ) In the nucleus

Proton 1 +1 ( positive ) In the nucleus

Electron 1 / 1850 -1 ( negative ) Arranged in energy levels or shells around the


• The sub-atomic components of atoms.

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• Atomic Number – no. of protons in the atom’s Hydrogen has 1 proton – atomic no. is 1. Carbon has 6 protons – atomic no. is 6.

• Mass Number – total no. of protons and neutrons in Lithium has 3 protons & 4 neutrons – mass no. 7

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Relative Atomic Mass

The mass of atom in relation to the mass of a reference atom.

Element chosen on which to relate mass of all atoms is Carbon.

Eg. Carbon (12) - 6 protons + 6 neutronsHydrogen - 1 proton + 0 neutron = mass no. 1

*so relative atomic mass for Hydrogen is one twelfth of Carbon.

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Isotopes• are atoms that have same atomic number but different mass numbers.• Atom gains / loses one or more neutrons from nucleus will become


• Are present in most elements and may also be man made.

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• From left to right across a period = protons/electrons count increases by one.

• Atomic number = increases from left to right.• Group 1. Vertical column2. Elements have same number of outer shell /

valency electrons.

Noble gases - group 8.Transition metals - middle of the table.Pure metals - group 1, except Hydrogen.

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• Atoms bond together to form a molecule.

Monatomic Molecule Chemical Compounds


Made up of two or more atoms.Consists of single type of atom.

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Structure of Atoms

• Electrons are arranged in energy levels and shells around the nucleus and with increasing distance from the nucleus .

• Valence electrons : - how many electrons an atom has in its outer shell.

- determines the chemical behavior.- The more valence electrons, the more want for the atom to get more electrons.

• Valency : 1. bonding capacity of an atom; usually equals the number of unpaired

electrons in the atoms outermost shell.2. No. of bonds it can make with a monovalent atom such as Hydrogen.


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The Molecule• A group of two or more similar or dissimilar atoms bonded together .

• Metallic bonds - metal + metal (eg: Alloy)- do not produce molecules- weakest bonds of all

• Ionic bonds - metal + non-metal (eg: sodium chloride)

• Covalent bonds - non metal + non metal (eg : water) - produce molecules, strongest bond.

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Chemical compounds• Defined as :‘ A substance made up of different elements that are chemically bonded and are so united that the whole has properties of its own, which are unlike those of its constituents.’

• All other molecules made up of two or more atoms are known as Chemical compound.

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States of matter

• All matter exists in one of three physical states.

• Physical state refers to the condition of a compound and has no

affect on a compound's chemical structure.• Solid. A solid has definite mass, volume and shape.

• Liquid. A liquid has definite mass and volume but takes the shape

of its container.

• Gas. A gas has definite mass but takes the volume and shape of

its container.

• increase in the temperature will increase the energy of its


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Changes between states

• Solid to Liquid – Fusioni. Heat energy required to complete process of converting unit

mass of substance from solid to liquid state without change of temperature is called the Latent Heat of Fusion.

• Liquid to Gas – Vaporisationi. Heat energy required to vaporise a unit mass of liquid without

temperature rise is called Latent Heat of Vaporisation.• Gas to Liquid – Condensation i. Heat energy expelled is called the Latent Heat of Condensation.• Liquid to Solid – Solidification i. Loss of molecular energy is the Latent Heat of Solidification.

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• Sublimationi. Some solid substance when heated do not melt, but form a


ii. Solid to vapour, without passing through liquid state.

iii. Eg. Dry ice

• Evaporation

i. Liquid change to vapour.

ii. Some liquid have low boiling point, so easily change from liquid

to vapour.

iii. Eg. Methylated spirits.

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Mechanics : statics ; -

forcescentre of gravitystress and strain

properties of matterpressure and buoyancy in liquids

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Newton’s law of motions

First Law of MotionA body at rest will remain at rest unless given an external force, or a body which is moving will keep on moving unless given an external force. ( Inertia )

Second Law of MotionA force proportional to the rate of change of its velocity is produced whenever a body ( or mass ) is accelerated.

F = ma

Third Law of MotionFor every action, there is an equal and opposite direction.

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STATICForces• If a Force is applied to a body it will cause that body to move in the

direction of the applied force.• force has both magnitude (size) and direction.• Forces cannot be directly observed, only their effects can be seen.

Compressive - force applied on an object to squeeze it.Tensile - pair of forces try to extend an object.Shear - Pair of forces tries to cause one face of material to

slide relative to an adjacent face.• Statics is used to describe study of bodies at rest when forces are


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• External force – force applied to an object from outside its boundaries.

• Internal force – force induced in the object to react against externally applied force. ( Reaction )

VECTOR AND SCALAR• Scalar – just a magnitude, there is no direction associated.

eg. Time , mass, volume.

• Vector – require both magnitude and direction to be fully Force, velocity, acceleration.

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Adding Forces• 2 or more forces act at same line, produce resultant force.• If at straight line, subtract or addition.

• Forces do not act in a straight line – use the ‘parallelogram law’.

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Worked example

Find the resultant of two forces of 4.0 N and 5.0 N acting at an angle of 45 degree 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 = 45 degree, see figure 8.

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Worked example: three forces acting on a mass.

First resolve each force into its vertical and horizontal components.

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MOMENTS AND COUPLES• If a body free to rotate about some point ;i. Applied force will cause rotationii. Force required dependent on how far from hinge force is applied. iii. Turning effect = magnitude and distance.

Moment = force x distance.

• In SI units, Newtons metres = Newton x metres

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The Principle of Moments

‘If a body is at rest under the action of several forces, the sum of the clockwise moments about any axis is equal to the sum of the anti clockwise moments about the same axis.’

Type 1 – beam balances where arms are of equal length.Type 2 – lever arrangement can best be seen in design of a wheelbarrow.Type 3 – large effort moves through small distance to overcome small load, which moves through a large distance.

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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 mmOC = 20 mmBC = 20 mm

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CENTRE OF GRAVITY • Gravity is a force which is always present and is a pulling force in

the direction of the center of the earth.

• This force acts on every body through an imaginary point called

the center of gravity (C of G).

• A point where all the weight of a body appears to be

concentrated. (total weight can be considered to act through that

datum position )

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(total weight can be considered to act through that datum position )

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• There must be a datum point, such as where when moment in clockwise

direction will be balanced by moment in anti-clockwise direction, if given

the same amount of force.

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Stability / balancing

• The lower the C of G, the stable an object is.• The wider the base, the more stable an object

is – C of G towards the base.

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• The location of the center of gravity in the

human body varies slightly, depending on

body shape.

• a male with a muscular upper body and

small hips, the center of gravity is higher

than in a female with narrow shoulders

and wide hips

• in an infant with a large head in relation

to its body, it is higher than in an adult

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• When force applied to C of G, the body will not rotate.

• But if the force is applied offset of the C of G, the body will rotate, or torque will produced.

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Expressed as an algebraic formula,W1 X1 + W2 X2 + W3 X3 = (W1 + W2 + W3) x G

Where G is the position of the centroid, with respect to the datum.G =

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C of g of an aircraft

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Why do we need to know c of g of an aircraft ?

• To ensure the aircraft is safe to fly, the center-of-gravity must fall within

specified limits established by the manufacturer.

• C of G range – C of G limits are specified longitudinal (forward and aft) and/or

lateral (left and right) limits within which the aircraft's center of gravity must

be located during flight.

• To evenly load the aircraft – equipments, passengers, baggage, cargo, fuel,


• So that C of G range will not be exceeded – prevent aircraft unstable during


• Also affects C of G in flight – fuel usage, passengers’ movement, etc.

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Similar to aircraft, force applied will be acted through the C of G, resulting in torque.

Aircraft rotate about its C of G.

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Stress • If force is exerted on a body, there will be mechanical pressure acting on

the body which is called the stress.

• A body with having twice the size of other body subjected to a force, it will

be stronger and less likely to fail due to applied the applied force.

• So, stress is said :

Stress =

*units : Newton metre -2 , Nm-2

• Components will fail due to over-stressed, not over-loaded.

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• Eg. A tennis ball sealed from atmospheric pressure. So, as long as the

external forces acting on it does not exceed the internal forces, the ball will

maintain its shape.

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• Forces applied to the body will cause distortion of the body and change to the

material’s cross-sectional area ;

eg. Tensile Forces will cause elongation .

Compressive Force will cause reduction in dimension.

• Most material have elastic properties ( it will to return to its original shape after

the force is removed ) - provided forces does not exceed limit of elasticity.• There are 5 types of stress in mechanical bodies :

i. Tension ii. Compression iii. Torsioniv. Bendingv. Shear

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• force that tends to pull an object apart


• resistance to an external force that tries to push an object together.

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Torsion• Torsional stress is applied to a material when it is twisted.• Torsion is actually a combination of both tension and compression

Bending• In flight, the force of lift tries to bend an aircraft's wing upward.

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Shear• combines tension and compression is the shear stress, which tries to slide

an object apart.

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Strain• 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.

Hooke’s law states that the amount of stretch (elongation) is proportional

to the applied force.

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How stress varies with stress when a steel wire is stretched until it breaks.

Hooke’s Law states that, ‘the amount of stretch (elongation) is proportional to the applied force.’

• Strain - the degree of distortion then has to be the actual distortion divided by the original length (in other words, elongation per unit length).

• Strain = change in dimension / original dimension ( No units )

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*for shearing and torsional stresses, they are expressed as ‘shearing action’ – when one layer of materials move relative to another in direction of applied force.

Shear strain – straight motion.Torsion strain – rotational motion.

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Compression strain

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Shear strain• when the applied load causes

one 'layer' of material to move relative to the adjacent layers.

Torsion strain• form of shear stress resulting

from a twisting action.

• Twist will be proportional to the applied torque.

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Shearing Strain

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Properties of matterDiffusion

• Is the spreading of a substance of its accord.

• due to molecular action, e.g. a smell, whether pleasant or not, travels

quickly from its source to your nostrils where it is detected.

• occurs in liquids and gases but not in solids.

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Surface Tension

• This suggests that the surface of a liquid behaves as if it is covered with an

elastic skin that is trying to shrink.

• 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 and prevents liquid

from spreading nag expanding out.

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Force of attraction between

molecules of same substances.


Force of attraction between molecules of

different substances.

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capillary• If a glass tube of small bore is dipped into water , the water rises u the

tube a few centimetres.

• 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.

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Mechanical properties of matterStrength

A strong material requires a strong force to break it

StiffnessA stiff material resists forces which try to change it’s shape or size.

ElasticityThe ability to recover to its original shape and size after the force deforming it has been reformed.

Ductility Materials that can be rolled into sheets, drawn into wires or worked into other useful shapes, without breaking are ductile.

Brittleness A material that is fragile and breaks easily .

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Pressure and buoyancy in liquid• The equivalent term associated with fluids is pressure:

pressure = force / area or p = F / A .

• Pressure is the internal reaction or resistance to that external force.

• Pascal’s Law : “pressure acts equally and in all directions throughout that fluid.”

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pressure can be transmitted to some other point in order to generate another force.

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Units of pressure

SI system pressure : Pascal = force per unit area ( Nm-2)

Atmospheric pressure :

Milibars ( mb ) or pounds per square inch ( psi )

Sea level standard atm presssure :

1013-2 mb or 14.69 psi ( at 0C)

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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.



1. The body will float--if the buoyancy is positive

2.The body will sink--if the buoyancy is negative

3.The body will be stuck--if the buoyancy is neutral

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Linear motion• Is the uniform motion in a straight line.• Motion is the change of position of a body with reference to

another body.

eg. A person sitting in a moving car and passes a building.

The person is considered to be at a state of rest in reference to the car.

The car is considered to be in motion in relation to the building.

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speed• Speed tells us how quickly an object is moving at any given point in time.

• Scalar quantity – does not take into account the direction of the object travelled.

• Average speed – dividing the distance travelled by the time taken.

Speed = rate of change of displacement or position=

v = where v represents speed.

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Worked example

• An aircraft flies at 80 km/ hr for 15 minutes and at 120 km/hr for a

further 15 minutes. How far has it travelled and what is the

average speed ?

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• Includes direction, distance in straight line and time.

• Vector quantity – magnitude + distance.

eg. An aircraft moves 200miles South West from A to B in one hour.

*the velocity is 200mph South West.

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acceleration• Is the rate of change of velocity over time.

• If the rate of change is constant, acceleration is described as being uniform.

• Acceleration - If the velocity of an object increases over time.

• Deceleration / retardation – If the velocity decreases over time.

• If a car is travelling at initial constant velocity ‘u’ and then accelerated with uniform acceleration ‘a’ to final velocity ‘v’, therefore :

a = ( v – u ) / tv = u + at

*Unit : m / s 2

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Equation of linear motionEquations of linear motion

V = u + at s = ½ ( u + v ) ts = ut + ½at2

v2 = u2 + 2as

Free falling objects • if a stationary object is released and free falls under attraction of the force

of gravity, it will accelerate at 9.81 m/s 2

• So the ‘a’ will be 9.81 m/s 2 , but in this case it is given the symbol ‘g’.• If the object is thrown upwards, its g will be – 9.81m/s2

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• Initial velocity of objects freefalling from rest is zero, so the equations for free fall are :

V = gt

S = ½ gt2

V2 = (2gs)2

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ROTATIONAL MOTIONCircular Motion• Rotational motion means motion involving curved paths and therefore

change of direction. • Only cases of constant acceleration are considered here.• They are equivalent to those linear equations of motion :

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Centripetal Force

• Continuous force applied to a body moving in circular path to keep it in

that particular circular path, preventing it from travelling in straight line

( tangential to circle it is rotating in ) – due to the inertia of the object.

• Basically a force acting inwards toward the centre of the circle.

• Is directly proportional to the mass of the object in circular motion.

• Is inversely proportional to the radius of the circle in which the object


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• Force = ma, so we can say that the object has an inwards acceleration called Centripetal Acceleration which is ;

a = v2 , so F =r

*where v is linear velocity and r is radius of circular path.

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Centrifugal Force• Is the equal but opposite reaction to the Centripetal Force. ( Newton’s 3rd Law )

• Tensile Force at the other end of the string acts outwards of the circle.

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Relationship between Angular and Linear Motion :

Providing that we know the value of the rotating body’s radius ( r ), we can use the relationship between the radius and the radian to convert angular values to linear values and vice versa.

Linear distance : rθLinear velocity : rωLinear acceleration : rα

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Periodic motion

• 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 motion.

• Example : pendulum

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• When the mass then displaced from its rest position, it will accelerate back towards

its rest position.

• On reaching it however, it will not stop, because its inertia carries it on to an equal

but opposite displacement.

• 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.

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Cycle –for one complete to and fro movement.

Periodic Time – time taken to complete 1 cycle.

Frequency – the number of cycle occurring in 1 seconds. ( Hertz – HZ )

Amplitude – maximum displacement of a body from its middle or rest position.

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• Periodic time, T can be calculated by ; T = 2 √ ( L / g )

Where ;L – length of the pendulumg – magnitude of acceleration due to gravity = 9.81 m/s2

For time period ( T ) and frequency ( f ) ;

T = 1 / f , f = 1 / T

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Spring – mass systems

• If the mass at the spring is displaced and the force is released, the spring

force will cause the mass to return to its original position.

• It will behave like pendulum, so it continue to move up and down before it

stops completely.

• Springs obey Hooke’s Law.

• The resulting motion, up and down, resulting in :

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Spring Force

Force ( F ) = mg = kE

if mg = kE, then the spring constant k = mg E

Also, extension E = mg k

Frequency (f) = ( 1 / 2 ) √ ( k/m )


F is force.m is suspended mass.g is the acceleration due to gravity.E is the spring extension.k is the spring constant.

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Simple theory of vibration, harmonics and resonance.

• Analysis of oscillating systems will show that they often obey simple but strict law.• Acceleration is proportional to the displacement from the neutral position, and in the

opposite sense to the direction of the velocity.• Referred to as Simple Harmonic Motion, when acceleration is directed towards fixed

point in its path and is proportional to its displacement from that point.

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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.

• Damped vibrations : Simple pendulum / spring mass will vibrate at constant frequency and

amplitude, once it is started. But vibrations will die away due to other motions such as friction, air

resistance, etc.

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• When force subjected to force vibration, it will vibrate along with natural

frequency of the object.

• If the natural frequency matches the resonant frequency with forced

vibration causes the amplitude to increase dramatically.

• Natural frequency tries to damped out the amplitude but cannot damp its

own natural frequency.

• So the two amplitudes combine to produce resonance.

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Velocity ratio, mechanical advantage and efficiency.

• Machine – utilise some form of motion to convert an applied force into a useful work output

• The input forces is often amplified many times by the machine so that we can overcome a heavy load with little effort.

• Eg ; levers, pulleys, gears, screws.

Velocity ratio

• Ratio of distance the effort is required to move in comparison with the distance the load moves in the same time.

Velocity Ratio = Distance moved by effort Distance moved by load

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Mechanical Advantage• Describes the ratio of load moved with the effort required to move it.

Mechanical Advantage = LoadEffort

*A rusty car jack will have a low mechanical advantage because much of the effort would be used to overcome the friction of a corroded screw thread.

Efficiency• Describes the ratio of the useful work done by a machine to the total work put into

it. Efficiency = Work Output x 100% or Mechanical Advantage x100%

Work Input Velocity Ratio• Expressed as percentage and is always less than 100%.

*Friction and slippage can detract from efficiency of a machine.

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LEVERS• Used to gain mechanical advantage.• Most basic form : seesaw that has weight at each end. ( weight on one end tends to rotate it clockwise, weight on the other end tends to rotate it anti-clockwise )

First Class Lever

• Lever has fulcrum between load and effort.• Less effort required to lift the load.

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Second Class Lever• Has fulcrum at one end of the lever and effort is applied to the opposite


Third Class Lever• Force is applied between fulcrum and load.• Used to move the load a greater distance than effort applied.• Disadvantages : much greater effort required to produce moment.

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• Pulley wheel has circumferential groove to accept a rope.

• Effort is applied by pulling on a rope.

• Tension created in the rope and movement are transmitted through

arrangement to the load.

• Single fixed pulley is a convenient means of lifting a light load.

• So, for instance, if one end of the rope is attached to a fixed object, pulling on

the other end will apply a doubled force to any object attached to the axle.

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Mass• Is the quantity of matter that it contains.• Constant regardless of its location. • Basic SI unit : kilogram (kg)• Imperial : pounds (lbs)

Weight• Force with which gravity attracts a body.• Varies with distance between body and centre of the earth, so if farther

than centre of the earth, the less it weighs.• So, it is said that an object in deep space does not has no weight, but do

has mass.• Gravitational acceleration is considered as 9.81m/s2

Force • Is a vector quantity that has magnitude, direction and a point of

application.• Which changes a body’s state of rest or of uniform motion.

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Inertia• Newton’s First Law• A body at rest will stay at rest unless given an external force, or a moving

body will continue on moving unless given and external force.• Is the resistance to movement or changes.

Work• Work is done when a force move.

Work done = force x distance moved in the direction of the force.• Unit : Newton metre (Nm) or the joule

1 joule = the work done when a force of 1 Newton is applied through a distance of 1 metre

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Power • The amount of work done in specific time.

Power = work done = force x distance time taken time

• Si unit : Watt ( W ) – 1 Joule/ second• Is the rate of work done when 1 Joule is achieved in one second.*One horsepower is the equivalent of 746 Watts

Brake Horse Power • To rate the engine power.• The engine is made to do work on a device known as dynamometer or ‘brake’ –

loads the engine output.

Shaft Horse Power• Measure output shaft of a turboprop engine, since the power produced at shaft

is what will be delivered to propeller.

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• Is the capacity to do work.

• Si unit : Joules

• Can appear as several form ; mechanical, chemical, heat, electrical, and


• But only deal with Mechanical Energy, and appears as potential and

kinetic energy.

• Energy cannot be created or destroyed, it can only be changed from one

form to another.

• Energy will not be converted into 100% work, they will always appear in a

less useful form such as wasted heat.

• But it may be converted into 100% energy.

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Potential Energy (PE)

• Energy is possesses by virtue of its position or state.

• Example : A mass raised to a height above the ground has potential energy

since its weight is capable of doing work as it descends to the ground

under the influence of gravity.

Work = force x distance

*since PE is for mass at elevated position,so distance is the height above


Potential Energy = mass x gravity x height

= mgh

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Kinetic Energy

• Energy it possesses by virtue of its motion.

• Arises from the work done on it.

• When body set in motion by a force doing work it acquires kinetic energy,

which will work against any forces that try to resist it.

Kinetic Energy = ½ mv2

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Conservation of Energy

“ Whenever energy is converted from one form to another,

none of it is lost.”

• The sum of the energy can always be accounted for in the other forms of

energy that may have converted into.

• After an object hits ground on a fall, the energy has converted into heat

energy and will be dissipated into the air.

• So a comparison between work out and work in is obviously a measure of

the system efficiency.

Efficiency = work output

work input

*usually expressed as percentage ( less than 100 % ).

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• Defined as energy between two bodies because of difference in


• If two bodies at different temperature, are bought into contact, their

temperature become equal.

• Energy that flows from a hot place to a cooler place.

• Heat energy can be transferred by ; conduction, convection and radiation.

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momentum• Product of its mass and velocity.

momentum = mass x velocity

• SI unit ; kgm/s

Impulse of a force• If a body was subjected to a sudden blow, shock load or impact, it will be

possible to measure change in momentum.• Forces which have the short time duration are called the Impulsive forces. • Change of momentum due to impulsive force is called the Impulse.• Impact duration small – impulsive force large. Impact duration large – impulsive force small.

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• Product of force and time or change in momentum.

• For example, if we rest the hammer, on top of the head of a nail, neither

the hammer nor the nail has any momentum.

• However, if we bring the hammer down from a height and strike the nail

sharply on its head, both the hammer and nail will move after the impact.

In a short time they will come to rest with the nail having penetrated

whatever it is under it.

• Impulse = change in momentum = mass x change in velocity.

= force ( N ) x time ( seconds )

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“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.”

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First law Mass remains at rest or continue to move at constant velocity, unless acted on by an external force.

First LawMass remains at rest or continue to move at constant velocity, unless acted on by an external force.

Second Law F = ma

Second LawRate of change of momentum is proportional to the applied force.

Third Law For every action, there is a n equal and opposite reaction.

Third LawIf mass A exerts force on B, then B exerts an equal but opposite force on A.

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Changes in Momentum

Change of Momentum = final momentum - initial momentum = ( m v ) - ( m u )

Rate of change of momentum = change in momentum time taken

= m v – m u t

*The rate of change of momentum is proportional to magnitude of force causing it.

• When two bodies, one of low mass and the other of high mass, are acted

upon by the same force for the same time, the low body mass will build up

higher velocity than the heavy mass.

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For example;

At (a) – mass A overtakes mass B.

At (b) – mass B will be accelerated by impulsive force delivered by mass A,

and mass A will be decelerated by an impulsive force delivered by B.At (c) – after the impact, mass A and B will have new velocities Va and Vb.

Momentum before impact equals to momentum after impact.

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Example :

A moving snooker balls, each ball has its own momentum before collision. After collision,

the sum of the two balls’ momentum will be the same as the sum prior to the collision

even though their velocities may have changed.

sum of momentums of balls = sum of momentum after impactprior to impact

mu1 + mu2 = mv1 + mv2

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Moment of inertia

• considers the effect of mass on bodies whose moment is rotational.

• Moment of inertia is a function of mass and radius.

• Consider the two cylinders, of equal mass, but different dimensions, capable of being rotated.

• The LH cylinder is easier to rotate than RH cylinder.

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• is a rotor having freedom of motion in one or more planes at right angles

to the plane of rotation.

• used in several of an aircraft’s instruments, which are vital to the safety of

the aircraft in bad weather.

• For example wheels, engines, propellers, electric motors and many other

components must run with perfect smoothness.

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• With the rotor spinning, the gyroscope will possess two fundamental


Gyroscopic rigidity or inertia

Gyroscopic precession

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Gyroscopic rigidity

• maintains the axis of rotation constant in space.

• 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.

• The degree to which the rotor offers resistance depends on 3 things :

Mass of the rotor – greater the mass, greater resistance to

change in direction of plane of rotation.

Angular speed of the rotor – higher the speed, greater the resistance.

Radius of gyration of the rotor – bigger the radius, greater the rigidity.

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Gyroscopic Precession

• 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.

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I. The rotor will rotate about axis AA.II. Apply a force so that it acts on the rim of the rotor at 900.III. Move this force around the rim of the rotor so that it moves through

900 and in the same direction as the rotor spins.IV. Precession will move the rotor in the direction that will result in the

axes of applied force and of rotation coinciding.V. For a constant gyroscopic speed, the rate of precession is proportional

to the applied force. VI. The opposite also applies, so for a given force the rate of precession is

inversely proportional to rotor speed.

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Attitude indicator.

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Determining Precession Direction

• If a mass is mounted on a rotating shaft, and the centroid is offset from the axis of rotation, mass will exert centrifugal force on the shaft.

• Even if the eccentricity is small, force may be considerable at high speed.

• So, it will cause the shaft to bend.• If large stresses produced, will cause damage to bearing.• Addition by vibrations from supports and surroundings will cause

undesirable effect.• Some eccentricity is due to manufacturing imperfections or design, so it is

unavoidable.• Balancing is needed to eliminate effect of centrifugal force.• Eg ; weights put on car wheels to balance them, make it easier to drive at

high speed.

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• Force that resists any sliding movement between two contacting surfaces.

• Can be taken as advantage, for brakes on vehicle to try to walk on smooth


• Frictional force depends on nature between two surfaces.

• Acts in any directions but always acts in opposing motion.

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• angle of the plane (θ) is increased, the body remains stationary, until at some particular value of θ, it begins to move down the plane.

• At this maximum value, the force opposing motion Fmax = mg sin θ ,

• and the normal reaction between the body and the planeR = mg cos θ.F/ R = mg sin θ / mg cos θ = tan θ

• ratio F/R (tan θ) is termed the Coefficient of Friction ( μ ) - < 1.μ = F = tan θ


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Coefficient of Static Friction

• Static friction is friction between two solid objects that are not moving

relative to each other.

• The static friction force must be overcome by an applied force before an

object can move.

• The maximum possible friction force between two surfaces before sliding

begins is the product of the coefficient of static friction.

• sometimes referred to as limiting friction.• Coefficient of Static Friction (μ) = Friction Force ( F )

Normal Reaction ( N )• Frictional Force , F = μN

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Coefficient of Dynamic Friction

• Dynamic friction is when two objects are moving relative to each other

and rub together (like a sled on the ground).

• The amount of force required to keep the object moving is called the

coefficient of dynamic friction.

• usually less than the coefficient of static friction for the same materials

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There are several types of friction:

• Dry friction resists relative lateral motion of two solid surfaces in contact.

Dry friction is subdivided into static friction between non-moving surfaces,

and kinetic friction between moving surfaces.

• Fluid friction describes the friction between layers within a viscous fluid

that are moving relative to each other.

• Lubricated friction is a case of fluid friction where a fluid separates two

solid surfaces.

• Skin friction is a component of drag, the force resisting the motion of a

solid body through a fluid.

• Internal friction is the force resisting motion between the elements

making up a solid material while it undergoes deformation.

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Fluid dynamics

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• Fluid – term used for liquid and gases.

Specific gravity and density

• Density – mass per unit volume.• Varies with :

Temperature ( for solids and liquid )

Temperature and pressure ( gas )

Density ( ρ ) = mass volume

• Eg : liquid that fills a certain container has a mass of 756 kg. The container is 1.6 m long, 1.0 m wide and 0.75 of a metre deep. The liquid density should be ?

ρ = 756 = 630kgm-3


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• Standard conditions for the measurement of gas density is established at

00C and a pressure of 1013.25 milli-bars

(Standard atmospheric pressure).

• Temperature change will not change the mass of the substance, but as

temperature change, the substance tend to expand or contract, altering

the volume.

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Relative Density

• It is necessary to compare density of one substance with the other to

achieve a standard which all other substance can be compared.

• For solid and liquid, compare with water at 40C.

• For gas, compare with air.

Relative Density = mass of any volume of a substance

mass of equal volume of water / air

• Eg ; if a hydraulic fluid has relative density of 0.8, then 1L of the liquid

wieghs 0.8 times as much as 1L of water.

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• Used to measure the relative density of liquids.

• Glass float contained within cylindrical glass body.

• Weight at the bottom, scale at the top.

• When liquid drawn into the body, float displays relative density on

graduated scale.

• Immersion in pure water will give out reading of 1.000.

• App. in aviation : to measure battery electrolyte and fuel.

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• Measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress.

• Also known as "thickness" or "internal friction“.• Eg : water has ‘thin’ velocity, while honey has ‘thick’ velocity.• 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,

when it is great, it makes the flow of the liquid very slow.• Viscosity of a liquid rapidly decreases as its temperature rises.

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• Viscosity of different liquids can be compared in different ways.

• Eg ; if we allow fluids of different viscosity run out of container, the higher the viscosity, the longer time taken to empty the container.

• It is important to know about viscosity as aircraft uses fluids such as oil in the engine for lubrication.

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Fluid Resistance

Skin friction• Resistance present on a thin, flat plate which is edgewise on to a fluid flow.• Near the surface, the fluid is slowed up due to roughness of the skin, and

fluid can also be considered as stationary at the surface.• Effected by skin smoothness.• The rougher the skin, the higher the friction.

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Eddies or turbulent airflow• swirling of a fluid and the reverse current created when the fluid flows past an

obstacle.• The moving fluid creates a space devoid of downstream-flowing fluid on the

downstream side of the object.• Eg. If you put plate at right angles to flow, turbulence will be created behind

the plate and a very high resistance .

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Effect of streamlining• 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 laminar flow encounter obstructions, the streamline will break and

become irregular or turbulent.

If fluid flows slowly along pipe, the flow is streamline.

If flow is very fast and exceeds a certain critical speed, the flow will become turbulent.

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The Compressibility of Fluids

• All fluids are compressible, so that their density will change with pressure.

• Fluid ;

i. assume as incompressible – provided under steady flow

conditions, and changes of density small.

• Gases ;

i. easily compressed – except when changes of pressure and

density are small.

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Static and Dynamic Pressure

• Pressure acting on x x1 is due to the weight of the fluid acting downwards.• W = mg ( g = gravitational force ) mass = volume x density = height x cross-sectional area x density

= hAρ so, downwards force = h . ρ . g . A acting on A and pressure will be = h ρ g . A

A = h ρ g

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Static pressure

• Act depth h, within a stationary fluid of density ρ.

• Reduces density accompanied by reduced pressure.

Dynamic pressure

• Eg, moving air is essential in flight.

• Dynamic presssure = ½ ρv2 where ρ = density, v = velocity.

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Worked example

What is the pressure at 10.5 m deep of liquid in a pool having the density of 2.5kg/m3 ?

What is the depth of an object from the surface if pressure exerted on it is 100psi in a 5.3kg/m3 liquid ?

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Bernoulli’s Theorem• a principle that explains the relationship between potential and kinetic

energy in a fluid.• In a fluid the potential energy is that caused by the pressure of the fluid,

while the kinetic energy is that caused by the fluid’s movement.• 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.

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Laws Of Thermodynamics

First Law

Energy cannot be created or destroyed. It can only be converted from one to

another. When the energy converts from one form into other forms the total

quantity of energy remains the same.

Second Law

Heat can only transfer from a high temperature region to a lower temperature

region. It cannot naturally transfer the other way.

Third Law

The transfer of energy from matter becomes increasingly difficult as its

temperature approaches absolute zero. It is considered impossible at absolute


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Temperature• Heat is a form of energy that causes molecular agitation within a material. • is a measure of the kinetic energy of molecules.• Temperature scale;

Freezing – 00CBoiling – 1000C

• Farenheit Scale - when you increased the temperature of a gas by one degree Celsius, it expands by 1/273 of its original volume.

• 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 – absolute zero.

• On the Celsius scale absolute zero is - 2730C. On the Fahrenheit scale it is – 4600F.

• Conversion ;– °C x 9/5 + 32 = °F ( from Deg. Centigrade to Farenheit )– (°F - 32) x 5/9 = °C ( from Farenheit to Deg. Centigrade )

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Heat• is the exchange of thermal energy from a hot body to a cold body.• When a hot body and a cold body have contact, heat will flow from the

hot body to the cold body until they both reach thermal equilibrium (they are at the same temperature).

• For example ; When an aircraft’s brakes are applied, the kinetic energy of the moving aircraft is changed into heat energy by the rubbing action of the brake friction material against the brake discs.• SI system :

Joule (J)British thermal unit (Btu)calorie (cal)

Heat & Work Conversion Factors 1 J 0.2388 cal 1 cal 4.1868 J 1 Btu 1055 J 1 J 0.000 947 Btu 1 Btu 0.252 cal 1 cal 3.968

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Heat Capacity

• specific heat - amount of heat per unit mass required to raise the

temperature by one degree Celsius.

• Relationship between heat and temperature change :

Q = cmΔT

Q is the heat added in 0C

C the specific heat capacity in J / kg0C

M is the mass in kg

ΔT is Tfinal – Tinitial in 0C

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• Different materials require differing amounts of heat energy to change

their temperature.

• The heat energy required to change the temperature of 1 kg of material by

1 K is known as the specific heat capacity (c) of the material.

• Due to the high specific heat of water, oceans and large lakes serve as

temperature stabilisers.

• Land surfaces have a much lower specific heat, and the temperature can

vary significantly throughout the day.

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Heat Transfer

• three methods by which heat is transferred from one location to another

or from one substance to another, which are :




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• When body having high heat energy in contact with body having low heat


• Eg; When hot object in contact with cold object, energy of molecules from

hot object will be transferred to molecules of cold object until they have

the same amount of energy ( stabilize ).

• Various metals have different rates of conduction.

• Liquids are poor conductors of heat in comparison with metals.

• .Gases are even worse conductors of heat than liquids.

• Insulators are materials that reduce or prevent heat conduction.

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• process by which heat is transferred by the movement of a heated fluid.

• Transfer of heat by convection is often hastened by the use of a ventilating

fan to move the air surrounding a hot object.

• Eg ; when heat is absorbed by a free-moving fluid, the fluid closest to the

heat source expands and its density decreases.

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• is the only form of energy transfer that does not require the presence of


• refers to the continual emission of energy from the surface of all bodies.

• This energy is known as radiant energy of which sunlight is a form.

• This is why you feel warm standing in front of a window whilst it is very

cold outside.Expansion and Contraction

• All materials expand and contract with a change in temperature.

gases which expand the greatest amount .

Solids and liquids expand much less than gases

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Volumetric Expansion

• Expansion – considered as change in length, change in area or change in volume.

• Different materials expand at different rates.

Expansion of Solids• Expansion is proportional to the increase in temperature to the original

dimension and depends on the actual material used.

L2 - L1 = L1 (θ2 - θ1)α

L2 and L1 are final and initial lengths,

θ2 and θ1 are final and initial temperatures

α is a material constant (coefficient of linear expansion).

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Expansion of Fluids

• Fluids expand more than solids.

• For gases, as volume and temperature changes are usually accompanied

by pressure changes.

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The law of Thermodynamics

• Thermodynamics is the study of the way that one does work with heat.

• Energy conservation limits the amount of work we can get out of a certain amount of heat.

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first law of thermodynamics • States that energy is conserved. • The change in internal energy of a system is equal to the heat added to the

system minus the work done by the system.

ΔU = Q – W

(Δ is the mathematical symbol for a change in a quantity) ΔU is the change in internal energy,

Q is the heat added to the system W is the work done by the system.

• ΔQ is positive if it is put into the system, negative if it is taken out of the system.

• ΔW is positive if the system does work on its surroundings and is negative if work is done on the system.

• The internal energy is the sum of the kinetic and potential energy of the atom and molecules that make up the system.

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second law of thermodynamics

• general principle which places constraints upon the direction of heat

transfer and the attainable efficiencies of heat engines.

• States that heat transfer will occur naturally of its own accord down the

temperature gradient.

• Heat will naturally flow from a hot region to a cool region but not the

other way around.

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• the basic sense of the principle :

Heat will not flow spontaneously from a cold object to a hot


You cannot create a heat engine which extracts heat and converts

it all to useful work.

There is a thermal bottleneck which constrains devices which

convert stored energy to heat and then use the heat to accomplish


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• Ideal gas - one in which all collisions between atoms or molecules are

perfectly elastic and in which there are no intermolecular attractive forces.

• a collection of perfectly hard spheres which collide but which otherwise do

not interact with each other.

• all the internal energy is in the form of kinetic energy and any change in

internal energy is accompanied by a change in temperature.

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From the Ideal Gas Law ;


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Boyle’s Law

If temperature is constant ;

P1V1 = P2V2

Charles’ Law

If pressure is constant ;


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Isothermal and Adiabatic Processes

Isothermal process - in which the temperature in a system remains constant.Adiabatic process - one where no heat is added to, or taken away from the system.

Heat engine

• heat engine is a system that performs the conversion of heat or thermal energy to mechanical work.

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Heat Engine Processes

• a useful process is the adiabatic process where no heat enters or leaves the system.

• The first law of thermodynamics with Q=0, i.e. heat = zero shows that all the change in internal energy is in the form of work done.

• internal energy is proportional to temperature, there is no change in the internal energy of the gas during an isothermal process.

• All the heat added to the system is used to do work.

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Engine Cycle

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Carnot Cycle• consisting of two isothermal processes and two adiabatic processes.• can be thought of as the most efficient heat engine cycle allowed by

physical laws.• the Carnot efficiency sets the limiting value on the fraction of the heat

which can be so used. • In order to approach the Carnot efficiency, the processes involved in the

heat engine cycle must be reversible and involve no change in energy available to do work.

• This means that the Carnot cycle is an idealisation, since no real engine processes are reversible and all real physical processes involve some increase in energy available to do work .

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Heat Flow to Hotter Region

• internal energy will not spontaneously flow from a hot region to a cold region.

• But if external force is given, heat can flow from cold region to hot region.• Usually this is done with the aid of a phase change, i.e., a refrigerant liquid

is forced to evaporate and extract energy from the cold area. . Then it is compressed and forced to condense in the hot area, dumping its heat of vaporisation into the hot area.

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Heat Pump

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Optics ( light )

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Speed Of Light

• is one form of transmission of Electro-magnetic energy.

• travels at high speed (about 3 x 108 metres per second) and in straight


• Can be ‘bent’ or reflected.

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Laws Of Reflection and Refraction

Reflection• Light can also be reflected.• Observation and measurement will show that ;

a. the incident and reflected rays lie in the same plane.b. the angle of incidence equals the angle of reflection.

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Plain and Curved Mirrors

• When you look in a mirror, you see a reflection, usually termed an image. • For example, if an object is viewed from two different angle, the reflected

rays :– appear to come from which corresponds to the image.– lies on the same normal to the mirror as the object.– appears the same distance behind the mirror as the object is in front.– Appears as the same size.

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• For mirror that is not plain, it may be curved, spherical and parabolic.• Incidence equals reflection - still holds, but the curved surface allows the

rays to be focused or dispersed.

FP is known as the focal length.Note the rays actually pass through F, and a real image can be formed.

FP is still the focal length, but the image is virtual.

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• The size of the image depends on the position of the object.• image may be smaller or larger.

• If the object is near to the mirror, the image will be far and larger from the mirror but in opposite plane.

• If the object is far from the mirror, the image will be nearer and small, between the position of object and mirror in opposite plane.

magnification = image heightobject height

• For spherical mirror, magnification = image distance object distance

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• Concave mirrors (e.g. shaving mirrors) give a magnified, erect (right way

up) image, if viewed from close-to.

• Convex mirrors (e.g. driving mirrors) give a smaller, erect image, but with a

wide field of view.

• Parabolic reflectors can focus a wide parallel beam. By placing the bulb at

the focus, they can produce a strong beam of light. (Conversely, they can

focus microwave signals when used as an aerial).

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• A submerged object is often seen at reduced depth.• This is because the ray has been ‘bent’ at the water / air boundary.

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Refraction Index

• Since the angle of incidence and refraction is not the same ;sine i = μ ( a constant ) sine r

• depends on the 2 mediums involved.μ = speed of light in medium 1 speed of light in medium 2

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Ray (1) has been refracted across the

boundary, but ray (2) has been internally

reflected at the boundary.

critical angle of incidence when the ray in the

denser medium does not emerge, but travels

along the boundary.

• sine C = 1 / μ exists

• Refraction is the basic principle which

explains the workings of prisms and lenses.

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Total Internal Refraction• Refraction at a denser medium, a beam of light is bent towards the normal

and, vice versa.

• APB – ray reflected away. • AP’D – angle of incidence increased, angle of refraction becomes 900 . • AP”C – Angle of incidence further increased , angle of refraction is >900 ,

remaining in the boundary.

• Total internal refraction – where none of lights passing through the boundary.

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Convex and Concave Lenses

• The light rays then meet the surface of the lens at an angle to the normal,

and are then refracted.

• Images can be real or virtual, erect or inverted, and larger or smaller.

• The nature of the image will depend on the type of lens, and the position

of the object in relation to the focal length of the lens, (the focal length is a

function of the curvature of the lens surfaces).

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Fibre Optics

• Depends upon the total internal reflection of light rays.

• Light can be trapped by total internal reflection inside a bent glass rod and

piped along a curved path as in the diagram below. a single, very thin

glass fiber behaves in the same way.

• They are small and so, once light is introduced into the fiber with an angle

within the confines of the numerical aperture of the fiber, it will continue

to reflect almost losslessly off the walls of the fiber and thus can travel

long distances in the fiber.

• If a bundle of parallel fibers is used to construct an optical transmission

line, images can be transferred from one point to another.

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Fibre Optic Imaging

• Principle : light striking at one end will be transmitted to the other end of

the fibre.

• If the arrangement of fibres in the bundle is kept constant then the

transmitted light forms a mosaic image of the light which struck the end of

the bundle.

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Wave motionand sound

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Wave motion

• For example, wave is created at a lake or pond due to disturbance, such as

a rock thrown into the water.

• The water wave has a crest and a trough and travels from one location to


• One crest is often followed by a second crest which is often followed by a

third crest, and so on.

• waves may be circular waves which originate from the point where the

disturbances occur; such circular waves travel across the surface of the

water in all directions.

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• Another example, if a slinky is stretched out from end to end, a wave can

be introduced into the slinky by either vibrating the first coil up and down

vertically or back and forth horizontally.

• As the wave moves along the slinky, each individual coil is seen to move

out of place and then return to its original position.

• The wave does not stop when it reaches the end of the slinky; rather it

seems to bounce off the end and head back from where it started.

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Categories of Waves• Waves come in many shapes and forms. • Some share basic characteristic properties and behaviours, some waves

can be distinguished from others based on some very observable (and some non-observable) characteristics.

• It is common to categorise waves based on these distinguishing characteristics.

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Direction Of MovementTransverse wave

• Wave moves in a direction perpendicular to the direction which the wave


• If pulse is applied to the left end by vibrating it up and down, the energy

will be transported from left to right and particle will be displaced upwards

and downwards.

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Longitudinal Wave

• Wave in which particles of the medium move in a direction parallel to the

direction which the wave moves.

• If force is applied to the left end by vibrating it left and right, the energy

will be transported from left to right and particle will be displaced

rightwards and leftwards.

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Ability to transmit through vacuum

Electromagnetic Wave• wave which is capable of transmitting its energy through a vacuum.• Electromagnetic waves are produced by the vibration of electrons within

atoms on the Sun's surface. • These waves subsequently travel through the vacuum of outer space,

subsequently reaching Earth.• Eg ; Light Wave.

Mechanical Wave • Wave which is not capable of transmitting its energy through a vacuum. • Require a medium in order to transport their energy from one location to

another.• Eg ; Sound Wave.

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Anatomy of Waves

• dashed line - equilibrium or rest position of the string.

(if there were no disturbance moving through it )

• Points A and F - crests of this wave (point on the medium which exhibits

the maximum amount of positive or upwards displacement from the rest


• Points D and I - troughs of this wave (point on the medium which exhibits

the maximum amount of negative or downwards displacement from the

rest position )

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• Amplitude of a wave - maximum amount of displacement of a particle on

the medium from its rest position. ( from rest to crest )

• Wavelength ;

a. Simply the length of one complete wave cycle.

b. Wave has a repeating pattern. And the length of one such

repetition (known as a wave cycle) is the wavelength.

c. Can be measured as the distance from crest to crest or from

trough to trough. ( B – G, E - J, D – I )

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Longitudinal wave

• is a wave in which the particles of the medium are displaced in a direction

parallel to the direction of energy transport.

• Wavelength - determined by measuring the distance between any two

corresponding points on adjacent waves. ( measure distance from a

compression to the next compression or from a rarefaction to the next

rarefaction ; A – C , B – D )

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Frequency and Period of a Wave

• Frequency ;a. refers to how often the particles of the medium vibrate when a

wave passes through the medium. b. number of complete vibration cycles of a medium per a given

amount of time and it as the units of cycles per second or Hertz (Hz) where 1 Hz is equivalent to 1 cycle/second.

c. . A detector could be used to detect the frequency of these pressure oscillations over a given period of time. d. unit : Hertz, Hz ( cycle / sec )

• Period ; a. the time which it takes to do something.b. the time for a particle on a medium to make one complete vibration cycle.c. When an event occurs repeatedly, then we say that the event is

periodic.d. measured in units of time such as seconds, hours, days or years.

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Interference Phenomena

Wave interference • phenomenon which occurs when two waves meet while travelling along

the same medium.

• Causes the medium to take on a shape which results from the net effect of the two individual waves upon the particles of the medium.

• Eg ; if 2 crests of wave having amplitude of +1 move into each other, the resulting sine crest +2 is created at the moment when the 2 waves overlapped.

• Sometimes called Constructive Interference.

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Constructive Interference

• Occurs at any location along the medium where the two interfering waves

have a displacement in the same direction.

• If both waves have an upward displacement; consequently, the medium

has an upward displacement which is greater than the displacement of the

two interfering pulses.

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Destructive Interference

• type of interference which occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction.

• If a sine crest with an amplitude of +1 unit meets a sine trough with an amplitude of -1 unit, destructive interference occurs.

• The two pulses cancel each other for the duration of the overlap.• Once the two pulses pass through each other, there is still a crest and a

trough heading in the same direction which they were heading before interference.

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Principle of Superposition“When two waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location.”

• To determine the shape of the resultant wave caused by the interference of two separate


• Standing Wave pattern :

An interference phenomena.

When the vibration frequency of the source causes reflected

waves from one end of the medium to interfere with incident

waves from the source in such a manner that specific points along

the medium appear to be standing still.

Only created within the medium at specific frequencies of

vibration; these frequencies are known as harmonic frequencies,

or merely harmonics.

From the presence of two waves (sometimes more) of the same

frequency with different directions of travel within the same


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• Anti-nodes - point A on the medium moves from a positive to a negative

displacement over time.

• Nodes - point B on the medium is a point which never moves.

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Standing Wave

• stationary wave - is a wave that remains in a constant position.

• often applied to a resonant mode of an extended vibrating object.

• created by constructive interference of two waves which travel in opposite

directions in the medium, but the visual effect is that of an entire system

moving in simple harmonic motion.

• modes of vibration associated with resonance in extended objects like

strings and air columns have characteristic patterns called standing waves.

• arise from the combination of reflection and interference such that the

reflected waves interfere constructively with the incident waves.

• An important part of the condition for this constructive interference for

stretched strings is the fact that the waves change phase upon reflection

from a fixed end.

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Sound• Sound wave - pressure disturbance which travels through a medium by

means of particle interaction. • As one particle becomes disturbed, it exerts a force on the next adjacent

particle, thus disturbing that particle from rest and transporting the energy through the medium.

• speed of a sound wave refers to how fast the disturbance is passed from particle to particle.

• frequency refers to the number of vibrations which an individual particle makes per unit of time.

• speed refers to the distance which the disturbance travels per unit of time.

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Speed of Sound

• Is determined to be 331 ½ m / s at 00C – 1087 ft /s , 741 mph or 644 kts.

• liquids are better transmitters of sound.

• Eg. sound waves travel approx. 4 times faster in water than in air and

speed of sound in solids is even greater, sound travels through steel is 15

times faster than it travels in air.

speed ( sound wave ) = frequency x wavelength

• Alteration in wavelength effects the frequency, but not the wave speed.

• Doubling of wavelength results in halving the frequency, but wave speed

doe not change.

• The speed of sound wave depends on the properties of the medium

through which it moves.

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• Primarily affected by temperature, the lower the temperature, the lower

the speed of sound.

speed of sound = √ ( γ R T )

where γ = ratio of specific heats of the gas

R = gas constant

T = gas temperature ( in Kelvin )

• It determines the nature and formation of shock waves.

Mach No = True Airspeed of aircraft

Speed Of Sound

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• amount of energy which is transported past a given area of the medium per unit of


• Often referred to as ‘loudness’ , the energy of the wave.

• The greater the amplitude of vibrations of the particles of the medium, the greater the

rate at which energy is transported through it, and the more intense that the sound

wave is.

• Intensity is the energy/time/area; and since the energy/time ratio is equivalent to the

quantity power, intensity is simply the power/area.

Intensity = Energy or Intensity = Power

Time x Area Area

• Eg ; if amplitude of sound wave is doubled, intensity will increase fourfold.

• Unit : Watts / Meter2 . But scale to measure intensity; Desibel (dB)

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Pitch• Best described as position on musical scale.

• The ears of humans (and other animals) are sensitive detectors capable of

detecting the fluctuations in air pressure which impinge upon the


• The human ear is capable of detecting sound waves with a wide range of

frequencies, ranging between approximately 20 Hz to 20 000 Hz.

• The sensations of these frequencies are commonly referred to as the


• A high pitch sound corresponds to a high frequency and a low pitch sound

corresponds to a low frequency.

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Doppler Effect

• effect produced by a moving source of waves in which there is an apparent

upward shift in frequency for the observer and the source are approaching

and an apparent downward shift in frequency when the observer and the

source is receding.

• Occur with all types of waves - most notably water waves, sound waves,

and light waves.

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• Doppler Effect observed because the distance between the source of sound and the observer is changing.

• If the source and the observer are approaching, then the distance is decreasing and if

the source and the observer are receding, then the distance is increasing.

• If the source is moving towards the observer, the observer perceives sound waves

reaching him or her at a more frequent rate (high pitch); and if the source is moving

away from the observer, the observer perceives sound waves reaching him or her at a

less frequent rate (low pitch).