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Physics by COBC - Issue 1 - 22 December, 2009 Page 1

Contents

1. SI UNITS ................................................................................................. 4

1.1 BASE UNITS................................................................................... 1-1

1.2 DERIVED UNITS.............................................................................. 1-2

2. FORCES ................................................................................................. 2-1

2.1 DEFINITION.................................................................................... 2-12.2 TRIANGLE OF FORCE...................................................................... 2-1

2.2.1 Graphical Method .......................................................... 2-1

2.3 POLYGON OF FORCE...................................................................... 2-2

2.4 ADDITION AND SUBTRACTION OF FORCES....................................... 2-3

2.5 VECTORS...................................................................................... 2-3

2.6 RESULTANTS................................................................................. 2-3

2.7 EQUILIBRIUMS............................................................................... 2-3

2.8 RESOLUTION................................................................................. 2-3

2.9 GRAPHICAL SOLUTIONS.................................................................. 2-4

2.10 SOLUTIONS BY CALCULATION......................................................... 2-5

2.11 MOMENTS AND COUPLES................................................................ 2-7

2.12 CENTRE OF GRAVITY...................................................................... 2-8

2.13 1STMOMENT OF AREA..................................................................... 2-11

2.14 STRESS......................................................................................... 2-12

2.15 STRAIN.......................................................................................... 2-12

2.16 SHEAR.......................................................................................... 2-13

2.17 TORSION....................................................................................... 2-13

2.18 SHEAR FORCE DIAGRAMS............................................................... 2-14

2.19 BENDING MOMENT DIAGRAMS......................................................... 2-163. ENERGY ................................................................................................. 3-1

3.1 WORK........................................................................................... 3-1

3.2 CONSERVATION OF ENERGY........................................................... 3-2

3.3 POWER......................................................................................... 3-2

3.4 MOMENTUM................................................................................... 3-2

3.5 CONSERVATION OF MOMENTUM...................................................... 3-2

3.6 CHANGES IN MOMENTUM................................................................ 3-2

3.7 IMPULSE OF A FORCE..................................................................... 3-3

3.8 INERTIA......................................................................................... 3-3

3.9 MOMENT OF INERTIA...................................................................... 3-3

3.10 2NDMOMENT OF AREA.................................................................... 3-3

4. GYROSCOPES ...................................................................................... 4-1

4.1 PRINCIPLES................................................................................... 4-1

4.2 RIGIDITY........................................................................................ 4-1

4.3 PRECESSION................................................................................. 4-1

4.4 TORQUE........................................................................................ 4-1

5. FRICTION ............................................................................................... 5-1

5.1 PRINCIPLES................................................................................... 5-1

5.2 FRICTION CALCULATION................................................................. 5-1

6. KINEMATICS .......................................................................................... 6-1


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6.1 PRINCIPLES................................................................................... 6-1

6.2 SPEED........................................................................................... 6-1

6.3 VELOCITY...................................................................................... 6-1

6.4 ACCELERATION !.

6.5 VECTORS...................................................................................... 6-2

6.6 LINEAR MOTION............................................................................. 6-3

6.7 DISTANCE-TIME GRAPH.................................................................. 6-3

6.8 VELOCITY TIME GRAPH................................................................... 6-3

6.9 AREA............................................................................................. 6-4

6.10 CONSTRUCTION AND USE OF EQUATIONS........................................ 6-5

7. ROTATIONAL MOTION ......................................................................... 7-1

7.1 CIRCULAR MOTION......................................................................... 7-1

7.2 CENTRIPETAL FORCE..................................................................... 7-1

7.3 CENTRIFUGAL FORCE..................................................................... 7-2

8. PERIODIC MOTION ............................................................................... 8-1

8.1 PENDULUM.................................................................................... 8-1

8.2 SPRINGMASS SYSTEMS............................................................... 8-1

9. HARMONIC MOTION ............................................................................. 9-1

10. VIBRATION THEORY ............................................................................ 10-1

11. FLUIDS ................................................................................................... 11-1

11.1 DENSITY........................................................................................ 11-1

11.2 SPECIFIC GRAVITY......................................................................... 11-1

11.3 BUOYANCY.................................................................................... 11-1

11.4 PRESSURE.................................................................................... 11-1

11.5 STATIC AND DYNAMIC PRESSURE.................................................... 11-3

11.6 ENERGY IN FLUID FLOWS................................................................ 11-3

12. HEAT 12-1

12.1 TEMPERATURE SCALES.................................................................. 12-1

12.2 CONVERSION................................................................................. 12-1

12.3 EXPANSION OF SOLIDS................................................................... 12-112.3.1 Linear ............................................................................. 12-112.3.2 Volumetric ...................................................................... 12-2

12.4 EXPANSION OF FLUIDS................................................................... 12-2

12.5 CHARLES LAW................................................................................ 12-2

12.6 SPECIFIC HEAT.............................................................................. 12-312.7 HEAT CAPACITY............................................................................. 12-3

12.8 LATENT HEAT /SENSIBLE HEAT....................................................... 12-3

12.9 HEAT TRANSFER............................................................................ 12-3

13. GASES .................................................................................................... 13-1

13.1 LAWS............................................................................................ 13-1

13.2 RATIO OF SPECIFIC HEATS.............................................................. 13-1

13.3 WORK DONE BY ,OR ON,A GAS...................................................... 13-2

14. LIGHT 14-1

14.1 SPEED OF LIGHT............................................................................ 14-114.2 REFLECTION.................................................................................. 14-1

14.3 PLAIN AND CURVED MIRRORS......................................................... 14-1


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14.4 REFRACTION................................................................................. 14-2

14.5 REFRACTIVE INDEX........................................................................ 14-3

14.6 CONVEX AND CONCAVE LENSES..................................................... 14-4

15. SOUND ................................................................................................... 15-1

15.1 SPEED OF SOUND.......................................................................... 15-1

15.2 FREQUENCY.................................................................................. 15-1

15.3 INTENSITY..................................................................................... 15-1

15.4 PITCH............................................................................................ 15-115.5 DOPPLER EFFECT.......................................................................... 15-1

16. MATTER ................................................................................................. 16-1

16.1 STATES OF MATTER....................................................................... 16-1

16.2 ATOMS.......................................................................................... 16-116.2.1 The Structure of an Atom .............................................. 16-116.2.2 The Fundamental Particles ............................................ 16-216.2.3 Particle Function ............................................................ 16-2

16.3 PERIODIC TABLE !.


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Physics by COBC - Issue 1 - 22 December, 2009 Page 1-1

1. SI UNITS

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. Mattercan 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 cancause a weight to be moved, or lifted.

As more and more 'happenings' have been studied, the subject of physics hasgrown, and physical laws have become established, usually being expressed interms of mathematical formula, and graphs. Physical laws are based on thebasicquantities - length,massand time,together with temperature andelectrical current. Physical laws also involve other quantities which are derivedfrom the basic quantities.

What are these units? Over the years, different nations have derived their ownunits (e.g. inches, pounds, minutes or centimetres, grams and seconds), but anInternationalSystem is now generally used - the SI system.

The SI system is based on the metre (m), kilogram (kg) and second (s) system.

1.1 BASE UNITSTo understand what is meant by the term derivedquantities or units considerthese examples; Areais calculated by multiplyinga length by another length,so the derived unit of area is metre2(m2). Speedis calculated by dividingdistance (length) by time , so the derived unit is metre/second (m/s). Accelerationis change of speed divided by time, so the derived unit is:

( ) second)persecondper(metres

mss

m2=

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)


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1.2 DERIVED UNITS

Some physical quantities have derived units which become rather complicated,and so are replaced with simple units created specifically to represent thephysical quantity. For example, force is mass multiplied by acceleration, which islogically kg.m/s2(kilogram metre per second per second), but this is replaced bythe 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-multiplesare 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' ()


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2. FORCES

If a Force is applied to a body, it will cause that body to move. A body that isalready moving will change its speed or direction. Note that the term 'change itsspeed or direction' implies that an acceleration has taken place.

This is usually summarised in the formula;

F = ma

Where F is the force, m = mass of body and a = acceleration.

The units of force should be kg.m/s2but this is replaced by the Newton.

2.1 DEFINITION

Hence, "A Newton is the unit of force that when applied to a mass of 1 kg. causesthat mass to accelerate at a rate of 1 m/s2.

Forces can also cause changes in shape or size of a body, which is importantwhen analysing the behaviour of materials.

2.2 TRIANGLE OF FORCE

Two or more forces can be added or subtracted to produce a Resultant Force.If two forces are equal but act in opposite directions, then obviously they canceleach other out, and so the resultant is said to be zero. Two forces can be addedor subtracted mathematically or graphically, and this procedure often produces aTriangle of Force.

Firstly, it is important to realise that a force has three important features;magnitude (size), direction and line of action.

Force is therefore a vectorquantity, and as such, it can be represented by anarrow, drawn to a scale representing magnitude and direction.

2.2.1 GRAPHICAL METHOD

Consider two forces A and B. Choose a starting point O and draw OA torepresent force A, in the direction of A. Then draw AB to represent force B.

The line OB represents the resultant of two forces.

Note that the line representing force B could have been drawn first, and force Adrawn second; the resultant would have been the same.



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Note how the two forces added together have formed 2 sides of the triangle; theresultant is the third side.

Also note that if a thirdforce, equalin lengthbut oppositein directionto theresultant is added to the resultant, it will cancel the effect of the two forces. Thisthird force would be termed the Equilabrant.

2.3 POLYGON OF FORCE

This topic just builds on the previous Triangle of Forces.

Consider three forces A, B and C. A and B can be added by drawing a triangle to

give a resultant.

If force C is joined to this resultant, a further or "new" resultant is created, which

represents the effect of all three forces.Now this procedure can be repeated many times; the effect is to produce aPolygon of Forces.

Again, the resultant can be derived mathematically. This will be considered in alater topic.



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Note that by drawing the right-angled triangle, with the single force F, and by

choosing angle relative to a datum, the two components become F sin and Fcos .

2.9 GRAPHICAL SOLUTIONS

This topic looks at deriving graphicalsolutions to problems involving the Additionof Vector Quantities.

Firstly, the quantities must be vector quantities. Secondly, they must all be thesame, i.e. all forces, or all velocities, etc. (they cannot be mixed-up).

Thirdly, a suitable scale representing the magnitudeof the vector quantity shouldbe selected.

Finally, before drawing a Polygon of vectors, a reference or datum directionshould be defined.

To derive a solution (i.e. a resultant), proceed to draw the lines representing thevectors (be careful to draw all lines with reference to the direction datum).

The resultant is determined by measuring the magnitude and direction of the linedrawn from the startpoint to the finishpoint.

Note that the orderin which the individual vectors are drawn is notimportant.



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2.10 SOLUTIONS BY CALCULATION

This topic achieves the same resultant as 2.9, but by mathematical methods.

Remember the topic dealing with Resolution (2.8). One vector was resolved intotwo mutually perpendicular components.

So if there are several vectors eachcan be resolved into two components.

e.g. F1 in direction 1, gives F1sin 1, and F1cos 1

F2 in direction 2, gives F2sin 2, and F2cos 2

F3 in direction 3, gives F3sin 3, and F3cos 3

etc, etc.

Once the components have been resolved, they can be added to give a totalforce in the Datum direction, and a total force perpendicular to the Datum.

These additions can be done laboriously 'by hand' but the modern scientificcalculator renders this unnecessary.

Each vector should be entered and multiplied by the cosine of its direction and

added consecutively to arrive at a total, F cos .

This procedure should be repeated, by multiplying each vector by the sine of its

direction, and added consecutively to give F sin .

To calculate the Magnitude of the resultant,

Add (F sin )2 + (F cos )2 (= F2)

And find the square root of the addition (=F)



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To calculate the Direction of the resultant,

Divide (F sin ) by (F cos ) (= Tan )

and find the Angle (direction) that has that resultant,

= tan-1 !

Note that the values of sine and cosine take both positive and negative values,depending on the direction.

The calculator automatically takes account of this during the procedure.

The only occasion when ambiguity can arise is when finding the angle of thedirection (there may be an error of 180). This can be resolved by inspection.

Note the following:

With reference to the ambiguity of direction,


!

!

note that = (A) gives the same angle (direction) as ! =

(B). Thus, F sin and F cos have to be inspected to see which is negative.Solution (A) or (B) can then be selected.

Similarly, !gives the same result as .

Again, inspect the values of F sin and F cos to see whether both are positiveor negative.


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2.11 MOMENTS AND COUPLES

In chapter 2.1, it was stated that if a force was applied to a body, it would move(accelerate) in the direction of the applied force.

Consider that the body cannot move from one place to another, but can rotate.The applied force will then cause a rotation. An example is a door. A forceapplied to the door cause it to open or close, rotating about the hinge-line. Butwhat 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.

So the turning effectof a force is a combination of the magnitude of the forceand its distance from the point of rotation. The turning effect is termed theMoment of a Force.

Moment (of a force) = Force x distance

In SI units, Newton metres = Newton x metres

Note: It is important to realise that the distance is perpendicular to the line ofaction of the force.

When several forces are concerned, equilibriumconcerns not just the forces,

but moments as well. If equilibrium exists, then clockwise (positive) moments arebalanced by anticlockwise (negative) moments.

When two equal but opposite forces are present, whose lines of action are notcoincident, then they cause a rotation.

Together, they are termed a Couple, and the moment of acouple is equal to the magnitude of a force F, multiplied bythe distance between them.



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2.12 CENTRE OF GRAVITY

Consider a body as an accumulation of many small masses (molecules), allsubject to gravitational attraction. The total weight, which is a force, is equal tothe 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, buthave different lines of action.

There must be datum position, such that the total moment to one side, causing aclockwise rotation, is balanced by a total moment, on the other side, whichcauses an anticlockwise rotation. In other words, the total weight can beconsidered to act through that datum position (= line of action).

If the body is considered in two different position, the weight acts through twolines of action, W1and W2and these interact at point G, which is termed theCentre of Gravity.

Hence, the Centre of Gravity is the point through which the Total Mass of thebody may be considered to act.



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For a 3-dimensional body, the centre of gravity can be determined practically byseveral methods, such as by measuring and equating moments, and thus is donewhen calculating Weight and Balance of aircraft.

A 2-dimensional body (one of negligible thickness) is termed a lamina, which onlyhas area (not volume). The point G is then termed a Centroid. If a lamina issuspended from point P, the centroid G will hang vertically below P1. Ifsuspended from P2 G will hang below P2. Position G is at the intersection asshown.

A regular lamina, such as a rectangle, has its centre of gravity at the intersectionof the diagonals.

Other regular shapes have their centre of gravity at known positions, see thetable below.



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A triangle has its centre of gravity at the intersection of the medians.



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The centre of gravity can also be deduced as shown.

If the lamina is composed of a several regular shapes, the centre of gravity of thatlamina can be deduced by splitting it into its regular sections, calculating themoments of these areas about a given datum, and then equating the sum ofthese moments to the moment of the composite lamina.

Expressed as an algebraic formula,

A, X, + A2X2 + A3X3 = (A1+ A2+ A3) x,

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

This is the principle behind Weight and Balance.

2.13 1ST

MOMENT OF AREA

In chapter 2.12, the last section introduced the formula;

A, X, + A2X2 + A3X3 = (ATotal) x,

The productArea x distance is termed the 1stmomentof area. Any datum (andassociated distanced) can be chosen, but once chosen, must be maintained aslong as the moments are being calculated.

The principle is used for Weight and Balance calculation as already stated, but 1stmoment of area is also important in other calculations, usually involving stressanalysis.



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2.14 STRESS

When an engineer designs a component or structure he needs to know whether itis strong enough to prevent failure due to the loads encountered in service. Heanalyses the externalforces and then deduces the forces or stressesthat areinduced internally.

Notice the introduction of the word stress. Obviously a component which is twicethe size in stronger and less likely to fail due an applied load. So an importantfactor to consider is not just force, but size as well. Hence stressis load divided

by area (size).


!(sigma) = (= Newtons per second metre).

Components fail due to being over-stressed, not over-loaded.

The external forces induce internal stresses which oppose or balance the

external forces.

Stresses can occur in differing forms, dependent on the manner of application ofthe external force.

Torsionalstress, due to twist, is a variation of shear.

Bendingstresses are a combination of tensile and compressive stresses.

All have the same SI unit i.e. N/m2.

2.15 STRAIN

If a length of elastic is pulled, it stretches. If the pull is increases, it stretch more;

if reduced, it contracts.Hookes law states that the amount of stretch (elongation) is proportional to theapplied force.

The degree of elongation or distortion has to be considered in relation to theoriginal length. The distortion is in fact a distortion of the crystal lattice.

The degree of distortion then has to be the actual distortion divided by theoriginal length (in other words, elongation per unit length). This is termed Strain.


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(epsilon) Strain = !

Note that strain has no units, it is a ratio and is then expressed as a percentage.

2.16 SHEAR

In chapter 2.15, different stresses were introduced, including shear stress.

Shearing occurs when the applied load causes one 'layer' of material to moverelative to the adjacent layers etc. etc.

Shear stressis still expressed as load/area but is usually represented by another

Greek symbol (tau).

Shear strain differs from direct strain. Whereas direct strain is expressed aschange in length / original length, shear strainis expressed in angular terms.

Shear strain = tangent of (gamma)

When a riveted joint is loaded, it is a shear stress and shear strain scenario.

The rivet is being loaded, ultimately failing as shown

2.17 TORSION

In chapter 2.15 Torsional stress was mentioned as a form of shear stressresulting 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 thesame effect and therefore the same unit as a Moment, i.e. Newton metres.



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If the bar is considered as a series of adjacent discs, what has happened is thateach disc has twisted, or moved relative to its neighbour, etc, etc. Hence, it is ashearing action.

The shear strain is equal to the angular deflection multiplied by radius r dividedby the overall length L,


! =

2.18 SHEAR FORCE DIAGRAMS

Engineers need to consider the effect of Shear Force and Bending momentswhen designing components and structures. These are often considered

graphically. In this topic, only simple, beam-type will be considered.

The beam AB is loaded with force F, and simply supported at A and B. It is inequilibrium. Hence, the forces and the moments balance.

Taking moments about A, Reaction at B, RBbalances the effect of F.

Hence RB. (a + b) = F.(a)

RB = F!

Similarly RA = F!

The effect of these forces is to create shearalong the beam.

But must be differentiated from .

The first diagram is defined as +ve shear, the second diagram as ve shear.

This can be shown on a shear force diagram (SFD).


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Considering the beam AB, the SFD is drawn as

Note that each changeto the SFD is equal to the load or force applied at thatposition or point. In this diagram only concentratedor point loads exist.

Now consider an uniformly distributed load. The SFD will look like this.

Note: Pointloads create BMDs that are triangular.

Distributedloads create BMDs that are curved.

(It is obviously possible to have loading patterns which are a combination ofboth).



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2.19 BENDING MOMENT DIAGRAMS

The diagram in 2.18 created shear forces, but the applied forces also createbending.

Consider a section x x at a distance x measured from end B. Then RBcreates amomentabout x x, which causes a bending effect.

The Bending moment = RB. x

As with shear force, moments causing have to be differentiated frommoments causing . The first case is +ve bending, the second case is vebending.

is commonly termed Sagging

is commonly termed Hogging.Note that as distance x increases, the Bending moment also increases.

Variations in Bending Moment are often shown on a Bending Moment Diagram(BMD).

Considering the beam AB, the BMD is drawn as;

Note that in this case, the BMD is all +ve (i.e. the beam is sagging everywhere)and note also that it increasesfrom zero as x increases to the leftof B, up to amaximum and then decreasesas the effect of RB is reducedby the effect ofF , finally becoming zero at A.



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(Note that BMD can be treated in the same way by considering distancesmeasured to the rightof A - the solutions are exactly the same).

A uniformly distributed load, whilst obeying the same principles, modifies theBMD.

As x increases left of B, RBcauses , but the distributed load also increases andcauses .

The BMD now looks like this.

Note: Pointloads create BMDs that are triangular.

Distributedloads create BMDs that are curved.



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!

! !

3.2 CONSERVATION OF ENERGY

This also suggests something most of us suspect there is no such thing as afree 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 isto be expected). So a comparison between work outand work inis obviously ameasure of the systems efficiency.

Efficiency =

It is usually expressed as a percentage, and so will clearly always be less than100%..

3.3 POWER

Recalling the man pushing the car, it was stated that the greater the distance thecar 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 becapable of pushing it through a certain distance within a certain time. A morepowerful manwill achieve the same distance in less time. So, the word Poweris introduced, which includes time in relation to doing work.

Power = Again, for simplicity and clarity, a dedicated unit of power has been created, theWatt.

The Watt is the Power output when one Joule is achieved in one second.

3.4 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 atry, used his momentum. This seems to suggest a combination of size(mass)and speedwere the contributing factors.

In fact, momentum = mass x velocity (mv).

3.5 CONSERVATION OF MOMENTUM

The principle of the Conservation of Momentum states:

When two or more masses act on each other, the totalmomentum of the masses remains constant, provided noexternal forces, such as friction, act.

Study of force and change in momentum lead to Newton defining his Laws ofMotion, which are fundamental to mechanical science.

The First law states a mass remains at rest, or continues to move at constantvelocity, unless acted on by an external force.

The Second law states that the rate of change of momentum is proportional tothe applied force.

The Third law states if mass A exerts a force on mass B, then B exerts an equalbut opposite force on A.

3.6 CHANGES IN MOMENTUM

What causes momentum to change? If the initial and final velocities of a massare u and v,

then changeof momentum = mv - mu

= m (v - u).


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Does the change of momentum happen slowly or quickly?


!

!

The rate of changeof momentum = m

Inspection of this shows that forceF (m.a) = m , so, a forcecauses achange in momentum.

The rate of change of momentum is proportional to the magnitude of the forcecausing it.

3.7 IMPULSE OF A FORCE

Note also that for a given force, the change of momentum will depend on the timeperiod, during which the force is applied. The product of force and time (Ft) istermed the impulse of a force.

3.8 INERTIA

Inertia is resistance to a change of momentum. We are familiar with this, e.g. aperson standing in a moving vehicle; if the vehicle stops, the person lurchesforward, as his mass contains momentum. The greater the mass, the greater willbe its inertia.

3.9 MOMENT OF INERTIA

Moment of Inertia considers the effect of mass on bodies whose moment isrotational.

This is important to engineers, because although vehicle move from on place toanother (i.e. the moment of the vehicle is translational) many of its componentsare rotating within it.

Consider two cylinders, of equal mass, but different dimensions, capable ofbeing rotated.

It will be easier (require less torque) to cause the LH cylinder to rotate. This isbecause 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 moredetailed study of the exact relationship is beyond the scope of this course, it canbe said that the M of I is proportional to the squareof the radius.

3.10 2ND

MOMENT OF AREA

In a previous topic, mention was made of the First moment of area (important fordetermining positions of centres of gravity) which was the product of Area xdistance.

The Secondmoment of area is an extension of this concept, and is proportional

to Area x distance2

.


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The second moment of area is important when calculating Moment of Inertia, andcan be done mathematically, although for simple shapes or areas, standardformulas exist to allow straight forward calculation.

Moment of Inertia (rather than mass) and Angular velocity combine together to

represent Angular Momentum (instead of mv, the symbols become )


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4. GYROSCOPES

4.1 PRINCIPLES

Gyroscopes are rotating masses (usually cylindrical in form) which aredeliberately employed because of the particular properties which theydemonstrate. (note, however, that any rotating mass may demonstrate theseproperties, albeit unintentionally).

Basic concepts can be gained by reference to a hand-held bicycle wheel.

Imagine the wheel to be stationary;it is easy to tiltthe axle one way or another.

4.2 RIGIDITY

Now rotatethe wheel. Because the massof the wheel is rotating, it now hasangular momentum. Two properties now become apparent.

The rotating wheel is now difficult to tilt, this is resistance is termed Rigidity.

4.3 PRECESSION

If sufficient force or torque is applied to tilt the wheel, the manner or direction inwhich it tilts or moves is interesting.

The movement of a gyroscope resulting from an applied torque is known asPrecession.

To calculate the manner or direction in which a gyroscope will precess, a simplerule applied.

Assuming the force is applied at A, then thegyroscope will behave as though the force hadbeen applied at a point B, 90 onward in thedirection of rotation.

4.4 TORQUE

The torque required to cause precession, or the rate of precession resulting fromapplied torque, depends on moment of inertia and angular velocity. Remember

that direction of rotation will determine direction of precession.



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5. FRICTION

5.1 PRINCIPLES

Friction is that phenomenon in nature that always seems to be present and actsso as to retard things that move, relative to things that are either stationary ormoving slowly. How large that frictional force is depends on the nature of the twosurfaces of the object concerned. Rough surfaces generally produce more

friction than smooth surfaces, and some materials are naturally 'slippery'.Frictioncan operate in any direction, but always acts in the sense opposingmotion.

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 valueof , it begins to move down the plane. This is because the frictional force (F)opposing motion has reached its maximum value.

5.2 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 .


!

!

= ! = tan

This ratio (tan ) is termed the Coefficient of Friction. It is generallyconsidered 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 staticfriction (friction associated withstationary objects) is greater than runningfriction (where objects are now inmotion).

A useful example is in flying-control systems, where engineers have to performboth static and running friction checks.


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!

!

!

!

6. KINEMATICS

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, accelerationhave been introduced, which do not refer to the force, but to the motion thatensues. Kinematics is the study of motion.

6.1 PRINCIPLES

When considering motion, it is important to define reference points or datums (ashas been done with other topics). With kinematics, we usually consider datumsinvolving positionand time. We then go on to consider the distance ordisplacement of the body from that position, with respect to time elapsed.

It is now necessary to define precisely some of the words used to describemotion, (which are common in everyday speech).

Distance and time do not need defining as such, but we have seen that they mustrelate to the datums. Distance and time are usually represented by symbolsx and t (although s is sometimes used instead of x).

6.2 SPEED

Speed = rate of change of displacement or position

=

v = ! where v represents speed.

A word of caution - this assumes that the speed is unchanging (constant). Ifnot, the speed is an averagespeed.

6.3 VELOCITY

Velocity is similar to speed, but not identical. The difference is that velocity

includes a directionalcomponent; hence velocityis a vector (magnitude anddirection - the magnitude component is speed).

Acceleration = rate of change of velocity

=

a = where a represents acceleration.

(In the above, v, represents the initial velocity, v2represents the final velocityduring time period t).

6.4 ACCELERATION

Note that as acceleration = rate of change of velocity, then it must also be avectorquantity. This fact is important when we consider circular motion, wheredirection is changing.

Remember, speed is a scalar, (magnitude only)Velocity is a vector (magnitude and direction)

If the final velocity v2is lessthan v1, then obviously the body has slowed. Thisimplies that the acceleration is negative. Other words such as decelerationorretardationmay be used.


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6.5 VECTORS

In exactly the same way as force vectors were added (either graphically ormathematically), so velocity vectors can be added. A good (aeronautical)example is the vector triangle used by pilots and navigators when allowing for theeffects of wind.

Here the pilot intended to fly from A to B (the vector AB represents the speed ofthe aircraft through the air), but while flying towards B the effect of the windvector BC was to 'blow' the aircraft off-course to C. So how is the pilot to fly to Binstead of C?

Obviously, the answer is to fly (head) towards D, so that the wind blows theaircraft 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 directionthe air speed (magnitude)

the track angle (direction)

The other two components may therefore be deduced, i.e. the aircraft headingand the aircraft groundspeed. Note that the differencebetween heading andtrack is termed drift. The aircraft groundspeed, (i.e. the speed relative to theground) is used to compute the travelling time.

This is a particular aeronautical example. More generally, if there are two vectorsv1and v2, then we can find relative velocity.

Note the difference in terminology and direction of the arrows. V2relative to v1means that to an observer moving at velocity V1, the object moving at velocity V2appearsto be moving at that relative velocity. (V

1relative to V

2is the apparent

movement of V1relative to V2).



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6.6 LINEAR MOTION

Linear motion means motion in a straight line, i.e. there is no change of direction.This type of motion is relatively easy to analyse and compute. Speed,acceleration and displacement can e deduced mathematically or graphically.

Graphical analysis introduces Distance / time and velocity / time graphs.

6.7 DISTANCE-TIME GRAPH

Velocity / time graphs are more versatile than distance / time graphs

although both may be considered.


!

! !

Speed = = slope of the graph. !

6.8 VELOCITY TIME GRAPH

Straight line = constant slope = constant speed.

(A straight line would clearly represent a stationarybody)

(Note - using symbols of calculus, v = !).

Acceleration a = = , this is usually re-arranged so v2 = v1+ at.


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(Note again - using symbols of calculus, a = != !).

The mathematical analysis of variableacceleration is beyond the scope of thiscourse, therefore the only graphs considered consist of straight lines only.

6.9 AREA


!

!

One importantconcept - velocity = ,

therefore distance = velocity x time

= area under the velocity - time graph.

Therefore, for a body changing velocity from v1to v2at constant rate ofacceleration.

Distance x = area = (v2+ v1)t + v1.t (triangle + rectangle)

x = !(v2+ v1)t


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or, (where v2 = v1 + at)


!

!

x = v1t + at2

6.10 CONSTRUCTION AND USE OF EQUATIONS

The equation of linear motion, assuming constant acceleration,

are v2 = v1 + at

x = (v1+ v2)tx = v1t + !at

2

(v1and x may appear as u and s, and x may appear as x2 x1 = difference

between position x2and position x1).

2axvvand 2122 +=


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7. ROTATIONAL MOTION

7.1 CIRCULAR MOTION

Rotational motion means motion involving curvedpaths and therefore change ofdirection. As with linear - motion, it may analysed mathematically or graphicallyand both types of motion are very similar in this respect, but employ differentsymbols. Again, only cases of constant acceleration are considered here, and

cases involving linear translation androtation are definitely ignored!

Firstly, consider the equation representing rotation. They are equivalent to thoselinear equations of motion.

Linear Rotational

v2 = v1+ at 2 = 1+ t

x = (v1+ v2)t 2 = (1+ 2)t

x = v1t + at2 = 1t + t

2

v2,2 = v2,1 + 2ax

2,2 = 2,1 + 2

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 employmeasurements in radians.

7.2 CENTRIPETAL FORCE

Consider a mass moving at a constant speed v, but following a circular path. Atone instant it is at position A and at a second instant at B.

Note that although the speedis unchanged, the direction, and hence thevelocity,has changed. If the velocity has changed then an accelerationmust

be present. If the mass has accelerated, then a forcemust be present to causethat acceleration. This is fundamentalto circular motion.

The acceleration present = !, where v is the (constant) speedand r isthe radius of the circular path.


!The forcecausing that acceleration is known as the Centripetal Force =

, and acts along the radius of the circular path, towards the centre.


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7.3 CENTRIFUGAL FORCE

More students are familiar with the term Centrifugal than the term Centripetal.What is the difference? Put simply, and recalling Newton's 3rdLaw, Centrifugal isthe equalbut opposite reactionto the Centripetal force.

This can be shown by a diagram, with a person holding a string tied to a masswhich 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 centrifugalreaction on person.

(Note again - cases involving changing speeds as well as direction are beyondthe scope of this course).


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8. 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 awayfrom, and then returned to its original position is known as the time periodandthe motion is known as periodic.

Many mechanisms or components behave in this manner - a good example is apendulum.

8.1 PENDULUM

If a pendulum is displaced from its stationary positionand released, it will swing back towards that position.On reaching it however, it will not stop, because itsinertia carries it on to an equal but oppositedisplacement. It then returns towards the stationaryposition, 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, withthe motion in the original direction

8.2 SPRING MASS SYSTEMS

If the mass is displaced from its original position andreleased, the force in the spring will act on the mass so as toreturn 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 timeand will result ina typical graph, which is sinusoidal.



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!

9. HARMONIC MOTION

Analysis of oscillating systems such as the pendulum or the spring-mass willshow that they often obey simple but strict laws. For example, the instantaneous

accelerationis given by the term -2x.

a = = -2x

(This basically states that the acceleration is proportionalto the displacement

from the neutral (undisturbed) position, and in the oppositesense to the directionof the velocity)

The constant is the frequency of the oscillation. (The periodof the oscillation= !).

Such motion is often referred-to as Harmonic motion and analysis reveals thesinusoidal pattern of such motion (beyond the scope of this course).


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10. VIBRATION THEORY

Vibration Theory is based on the detailed analysis of vibrations and is essentiallymathematical, relying heavily on trigonometry and calculus, involving sinusoidalfunctions and differential equations.

The simple pendulum or spring-mass would according to basic theory, continueto vibrate at constant frequency and amplitude, once the vibration had beenstarted. In fact, the vibrations die away, due to other forces associated withmotion, such as friction, air resistance etc. This is termed a Dampedvibration.

If a disturbing force is re-applied periodically the vibrations can be maintainedindefinitely. The frequency (and to a lesser extent, the magnitude) of thisdisturbing force now becomes critical.

Depending on the frequency, the amplitude of vibration may decayrapidly (adamping effect) but may growsignificantly.

This large increase in amplitude usually occurs when the frequency of thedisturbing force coincides with the natural frequency of the vibration of the system(or some harmonic). This phenomenon is known as Resonance. Designerscarry out tests to determine these frequencies, so that they can be avoided or

eliminated, as they can be very damaging.


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!

!

! !

11. FLUIDS

Fluid is a term that includes both gasesand liquids; they are both able to flow.

We will generally consider gases to be compressibleand liquids to beincompressible.

When considering fluids that flow, it is obvious that some flow more freely thanothers, or put another way, some encounter more resistance when attempting to

flow. Resistance to flow introduces the word Viscosity, highly viscous liquids donot flow freely. Gases generally have a low viscosity.

11.1 DENSITY

Density of a solid, liquid or gas is defined as = ! =

A largemass in a smallvolume means a high density, and vice versa. The unitof density depends on the units of mass and volume; e.g. density = kg/m3in SIunits.

Solids, particularly metals, often have a highdensity, gases are of lowdensity.

11.2 SPECIFIC GRAVITY

Density may be expressed in absoluteterms, e.g. mass per unit volume, or inrelativeterms; i.e. in comparison to some datum value. The datumwhich formsthe basis of Relative Density is the density of pure water, which id 1000 kg/m3.

Relative Density = .

Note that relative density has nounits, it is a ratio.

RD. = ! (often referred to a Specific Gravity)

The RD. of water is 1, and so substances with an RD. lessthan 1 floatin water;

with RD. greater than 1, they sink.

11.3 BUOYANCY

Buoyancy implies floatation, and may involve solids immersed in liquids or gases,one liquid in another, one gas in another and so on. It is a function of relativeDensities.

An object that floats has a R.D. less than the medium in which it floats. Itsweightis obviously supported by some interactive force (upthrust) between theobject and that medium.

Archimedes states that the upthrustis equal to the weightof the volume of the

medium that was displacedby the floating object, i.e. the volume of object belowthe surface.

11.4 PRESSURE

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 withsolids. The equivalent term associated with fluidsis pressure, so pressure =

. p = .


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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 thatexternal force. It is important to realise that pressure acts equally and in alldirections throughout that fluid. This can be very useful, because if a forceapplied at one point creates pressure within a fluid, that pressure can betransmitted to some other point in order to generate another force. This is theprinciple behind hydraulic (fluid) systems, where a mechanical input force drivesa pump, creating pressure which then acts within an actuator, so as to produce amechanical output force.


!

! !

In this diagram, a force F1is input to the fluid, creating pressure, equal tothroughout the fluid. This pressure acts on area A2, and hence an

output force F2is generated.

If the pressure P is constant, then = and if A2is greater thanA1, the output force F2is greater than F1.

A mechanical advantage has been created, just like using levers or pulleys. Thisis the principle behind the hydraulic jack.

But remember, you don't get something for nothing; energy in = energy out orwork in = work out, and work = force x distance. In other words, distancemovedby F1has to be greaterthan distance moved by F2.


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11.5 STATIC AND DYNAMIC PRESSURE

Static and Dynamic pressure.

In this diagram, the pressure acting on x x1 is due to the weight of the fluid (in thiscase 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 == 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 diagramdemonstrates. (we can "see" the liquid)

But the same principle applies to gases also, and we know that at altitude, thereduced density is accompanied by reduced static pressure.

We are not awareof the static pressure within the atmosphere which acts on ourbodies, 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 movingair isinvolved, as on a windy day for example. The pressure associated with movingair is termed dynamicpressure.

In aeronautics, moving air is essential to flight, and so dynamic pressure isfrequently referred-to.

Dynamic pressure = v2 where = density, v = velocity.

Note how the pressure is proportional to the squareof the air velocity.

11.6 ENERGY IN FLUID FLOWS

So the pressureenergy found in moving fluids, i.e. fluids that are flowing, has atleast two components, static and dynamic pressure. This is of fundamentalimportance when considering Theory of Flight.

(Note - if the fluid flow is not horizontal, then differences in potential energy, i.e.changes in "head" of pressure are theoretically present, but are generally ignoredwhen air is considered, because of its low density).


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12. HEAT

Heat is not the same as temperature(in a similar way that force is not the sameas pressure) but the two are often related. Heatis energy, while temperatureisrelatedto the kinetic energy(speed or movement) of the molecules of the bodyconsidered. But as heat transferis of importance in physics and engineering,and knowing that heat travels from hotto coldbodies, a means of assessingtemperaturebecomes essential.

12.1 TEMPERATURE SCALES

Several different temperature scales have evolved, the most common are theFahrenheit, Celsius and Kelvin scales. A diagram may be used to makecomparisons.

Boiling Point of water +212 100 373

(at standard pressure)

Melting ice +32 0 273

Fahrenheit Centigrade Kelvin

12.2 CONVERSION

An engineering student should be able to convert from one temperature toanother;

e.g. convert F to C - Subtract 32, then multiply by !

convert C to F - Multiply by !, then add 32

convert C to K - add 273

Example #1: Convert 20C to Fahrenheit.


!20 x + 32 = 36 + 32 = 68F

Example #2: Convert 15C to Kelvin

15 + 273 = 288K

Note also that when thermodynamic principles and calculations are considered, itis usually vital to perform these calculations using temperatures expressed inKelvin.

Note also that 0K is often termed absolute zero(it is the lowest temperaturetheoretically possible).

12.3 EXPANSION OF SOLIDS

Engineers are familiar with the effect of temperature on structures andcomponents, as the temperature increases, things expand(dimensionsincrease) and vice versa. Expansion effects solids, liquids and gases.

But how much does a component expand? The answer should be obvious.

Expansion is proportional to the increasein temperature to the originaldimension and depends on the actual material used.

12.3.1 LINEAR

So L2 - L1 = L1(2- 1)


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Where L2and L1are final and initial lengths,

2and 1are final and initial temperatures

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

12.3.2 VOLUMETRIC

Expansion can be considered as a change in length(see above), a change inareaor change in volume.

The change in volume, v2 - v1 = v1(2- 1)

Where = the coefficient of volumetric expansion. (note that = 3(seeabove)).

Different materialsexpand at different rates, and this may be used, forexample, when shrink fitting components

12.4 EXPANSION OF FLUIDS

Liquidsbehave in a similar way to solids when heated, but (a) they expand morethan solids, and (b) they expand volumetrically. Note that when heated, thecontainers tends to expand as well, which may or may not be important to a

designer.

Gaseshowever, behave in a rather more complex way, as volumeandtemperaturechanges are usually accompanied by pressurechanges.

12.5 CHARLES LAW

If a fixed mass of gas (e.g. air) is heated from temperature T1to T2, its initialvolume V1increases to V2. Note that the increase is linear (the graph follows astraight-line). Note that if the line is extended back, it crosses the T (x) axis at-273C, or absolute zero.


!The slopeis constant, therefore !is constant, or !=

(temperature must be expressed in K).

This illustrates Charles Law.

"The volume of a fixed mass of gas at constant pressure is proportional to theabsolute temperature".


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12.6 SPECIFIC HEAT

Heat is a form of energy, so how much heat is needed to increase thetemperature of a substance? Again, the answer is obvious.

The heat requireddepends on the temperature rise, the amount or massof thesubstance, and on the actual substancebeing heated. As a formula,

Q = mc (2- 1)

Where Q is the heat energy supplied (in joules), m is the mass (kg) 2and 1are

k.kgJin

final and initial temperatures, and c is the specific heatof the substance

considered .

12.7 HEAT CAPACITY

Here specificheat c is the amount of heat required to raise the temperature of1kg of a substance by 1K.

Heat capacity(C) of a body is the mass of the body x specific heat.

C = mc

12.8 LATENT HEAT / SENSIBLE HEAT

If water, for example, is heated at a constant rate, the temperature will rise,

shown by AB. At B, corresponding to 100C (the boiling point of water) the graphfollows BC, which represents the constant temperature of 100C. After a time,the graph resumes its original path, CD.

What was happened to the heat supplied during the time period between B andC?

The answer is that it was used, not to raise the temperature, but to changethestatefrom water into steam. This is termed latent heat, and also features whenice melts to become water.

So latentheat is the heat required to cause a changeof state, and sensible

heat is the heat required to cause a change of temperature.

12.9 HEAT TRANSFER

Heat transfer, where a substance at high temperature looses heat to a substanceat a lower temperature, is important as we may want it to occur, or not, as thecase may be.

Heat transfer may occur by conduction, convection or radiation.

Conductionis transfer of heat through the stationary substance.

Convectionis transfer by motion of the substance (this happens by fluid flow).

Radiationis by electro-magnetic radiation or wave propagation.



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For this module, the study is limited to conduction. Materials such as metals aregood conductors(e.g. silver, copper, aluminium) whilst other materials do notconduct readily and are termed insulators(e.g. wood, plastics, cork).

Note that there appears to be a similarity between thermal and electricalconduction or insulation.


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13. GASES

13.1 LAWS

Charles Law has already been referred to. Boyles law assumes constantvolume.

Boyle's Law states that "the pressure of a given mass of gas at constant

temperature is inversely proportional to its volume".

Summarised:


! !

! !

! !

is constant

is constant

pV is constant !

These laws are often combined to give:

is a constant

= mR where m is the massof gas considered, and R is the Gas Constant.

13.2 RATIO OF SPECIFIC HEATS

We have already defined the term specific heat as a quantity of heat supplied etc,and this is sufficient when considering solids and liquids. Gases can be adifferent case however, and the heat suppliedto produce a temperature rise willvary, depending on whether the gas is allowed to expand or not, whilst beingheated.

This leads to the twospecific heat values.

Cpis the specific heat of the gas which is maintained at constant pressure, butallowed to expand.

Cvis the specific heat of the gas which is maintained at a constant volume.

In the first case, the heat input raises the temperature, and causes the gas to

expand, during which the gas does work (gives out energy).

In the second case, the heat input only raises the temperature.


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!The ratio of the specific heats, symbol = , in which Cpis greater thanCv, hence !>1.

This particular relationship is frequently used in thermodynamics. pv = constant.

13.3 WORK DONE BY , OR ON, A GAS

Put simply, work is done bya gas that is expanding; work is done ona gas thatis being compressed.

This is a simplification but reference to a p.v. diagram is helpful.

The work done by or on the gas is given by the area under the p-v curve.

If we go from v1to v2(expansion) work is done by the gas.

If we go from v2to v1(compression) work is done on the gas.

The exactamount of work depends on the exact nature of the expansion /

compression, i.e. is the relevant gas law pv = constant or pvn= constant or pv

= constant?

These different equations give different curves, and hence different work values,but this is beyond the scope of this module.

Note also that an expanding gas tends to cool; a gas being compressed tends toheat-up.


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14. LIGHT

14.1 SPEED OF LIGHT

Light is one form of transmission of electro-magnetic energy. Light travels at highspeed (about 3 x 108metres per second) and in straight lines, although it can be'bent' or reflected.

14.2 REFLECTION

Light can also be reflected, usually by mirrors, which are made by depositing athin layer of metal on one side of a piece of glass. Some interesting facts may beobtained.

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.

14.3 PLAIN AND CURVED MIRRORS

When you look in a mirror, you see a reflection, usually termed an image. Thediagram above shows 2 reflected rays, viewing an object O from two differentangles. Note the reflectedrays appear to come from I which corresponds to the

image, and lies on the same normal to the mirror as the object, and appears thesame distance behindthe mirror as the objectis in front.

Note also that the image is a virtualimage, it can be seen, but cannot be shownon a screen.

Note also that it appears the same size as the object, and is laterally inverted.These are features of images in plane mirrors.

Mirrors can also be curves, sometimes they are spherical, sometimes parabolic.The basic law, incidence equals reflection - still holds, but the curved surfaceallows the rays to be focussed or dispersed.



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FP is known as the focal length.

Note the rays actually pass through F, and arealimage can be formed.

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

The sizeof the image depends on theposition of the object.

The image may be smaller or larger.

Magnification = !

(It can be shown for spherical mirrors that magnification = ! !.

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 widefield of view.

Parabolic reflectors can focus a wide parallel beam. By placing the bulb at thefocus, they can produce a strong beam of light. (Conversely, they can focusmicrowave signals when used as an aerial).

14.4 REFRACTION

Many people have noticed a strange optical phenomenon when looking atsubmerged objects. Such an object often appears to be at a reduced depth.



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The rays appear to have been bent at the water / air boundary. This is knownas Refraction.

14.5 REFRACTIVE INDEX

The angles of incidence and refraction are not equal, but they are related, shownas:


! = a constant =

is known as the refractive indexand depends on the 2 mediums involved.

It can be shown that = !

Another phenomena may occur. In the diagram, ray (1) has been refracted

across the boundary, but ray (2) has been internally reflectedat the boundary.

There is a critical angle of incidence when the ray in the denser medium does notemerge, but travels along the boundary.

The relationship sine C = !exists.Refractionis the basic principle which explains the workings of prisms andlenses.


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14.6 CONVEX AND CONCAVE LENSES

Lenses can be made of glass or plastic, and like mirrors, have spherical surfacesso as, to give concave or convex lenses. The light rays then meet the surface ofthe lens at an angle to the normal, and are then refracted. As the rays exist thelens, a second refraction takes place.

As with mirrors, images can be real or virtual, erect or inverted, and larger orsmaller. The nature of the image will depend on the typeof lens, and thepositionof the object in relation to the focal lengthof the lens, (the focal lengthis a function of the curvatureof the lens surfaces).


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15. SOUND

15.1 SPEED OF SOUND

Sound is transmitted by a wave motion that is unlike light or heat radiation, in thatit is not electro-magnetic, but relies on the transmission of pressure pulses - themolecules vibrate backwards and forwards about their mean position, and thisvibration transmits the pressure wave.

Sound travels much slower than light, only about 760 miles per hour at sea levelor 340 m/s.

The speed of sound is primarily affected by temperature, the lower thetemperature, the lower the speed of sound.

A formula exists, where;

speed of sound = ,RT

where = ratio of specific heats of the gas

R = gas constant

T = gas temperature (in Kelvin)

Speed of sound is of utmost importance in the study of aerodynamics, because itdetermines the nature and formation of shock waves. Because of this, aircraftspeed is often compressed in relation to the speed to sound.

! = Mach N

(Aircraft travelling at speeds greater than Mach 1 are supersonic, and generatingshock waves).

15.2 FREQUENCY

Frequency (f) of sound is related to the number of vibrations that the moleculesperform in a unit of time. The amount (or distance) which the molecules vibrateabout their main position is termed the amplitude.

Another term exists, i.e. wavelength (). A formula exists, linking frequency andwavelength.

f. = constant = speed of sound

High-pitches sounds are of high frequency, and vice versa.

15.3 INTENSITY

The intensity of sound (its 'loudness) is dependent on the intensity of the

pressure variations, and thus is related to the amplitude. The amplitude of thevibration is proportional to the energy input into the generation of the wave.

15.4 PITCH

Pitch is another word for frequency (see previous paragraph).

15.5 DOPPLER EFFECT

Doppler effect is the effect that is noticeable when for example, a car is heardspeeding towards the listener, then speeding away. The sound is initially at ahigh-pitch, which then becomes lower. This is because the sourceof the sound

(the car) is moving, which causes a change in the time interval betweensuccessive pressure variations in the ear of the listener (i.e. there appears to be achangein frequency, which is proportional to the speed of the car).


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16. MATTER

Matter is defined as anything that occupies space and may be classified in anumber of ways.

16.1 STATES OF MATTER

There are three normal states of matter:

Solid. A solid has definite mass, volume and shape.

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

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

16.2 ATOMS

If a water molecule could be magnified sufficiently it would be seen to consist ofthree smaller particles closely bound together. These three particles areATOMS, two of hydrogen and one of oxygen.

The water is a compound, the oxygen and hydrogen are elements. Everyelement has atoms of its own type. There are 92 naturally occurring elementsand therefore 92 types of naturally occurring atoms.

Every molecule consists of atoms. Molecules of elements contain atoms of thesame types, for example the hydrogen molecule consists of two atoms ofhydrogen joined together, the oxygen molecule consists of two atoms of oxygenjoined together, but the molecules of compound contain different atoms joinedtogether.

Most molecules contain more than one atom but some elements can exist assingle atoms. In such a case the atom is also the molecule. For example the

Helium atom is also the Helium molecule.An atom is the smallest indivisible particle of an element which can take part in achemical change.

16.2.1 THE STRUCTURE OF AN ATOM

The Nucleus and Electrons. Atoms themselves are also composed of evensmaller particles. Let us take an atom of hydrogen as an example. A hydrogenatom is very small indeed (about 10 10in diameter), but if it could be magnifiedsufficiently it would be seen to consist of a core or nucleus with a particle calledan electron travelling around it in an elliptical orbit.

The nucleus has a positive charge ofelectricity and the electron an equal negativecharge; thus the whole atom is electricallyneutral and the electrical attraction keepsthe electron circling the nucleus. Atoms ofother elements have more than one electrontravelling around the nucleus, the nucleuscontaining sufficient positive charges tobalance the number of electrons.

Protons and Neutrons. The particles in the nucleus each carrying a positivecharge are called protons. In addition to the protons the nucleus usually contains

electrically neutral particles called neutrons. Neutrons have the same mass asprotons whereas electrons are very much smaller only !of the mass ofa proton



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16.2.2 THE FUNDAMENTAL PARTICLES

Although other atomic particles are known, the three fundamental ones are:

Protons. The proton has unit mass and carries a unit positive charge.

Neutron. The neutron has unit mass but no electrical charge.

Electron. The electron has only unit of mass but it carries a unitnegative charge.

Thus although we have 92 types of naturally occurring atoms, they are all built-upfrom different numbers of these three fundamental particles.

Thus our picture of the structure of matter is as shown below.

16.2.3 PARTICLE FUNCTION

16.2.3.1 Protons

The number of protons in an atom determines the kind of material:

Eg. Hydrogen 1 proton

Helium 2 protonsLithium 3 protons

Beryllium 4 protons

etc

Copper 29 protonsetc

Uranium 92 protons

The number of protons is referred to as the atomic number, thus the atomicnumber of copper is 29.

16.2.3.2 Neutrons

The neutron simply adds to the weight of the nucleus and hence the atom. Thereis no simple rule for determining the number of neutrons in any atom. In factatoms of the same kind can contain different numbers of neutrons. For examplechlorine may contain 18 20 neutrons in its nucleus.

The atoms are chemically indistinguishable and are called isotopes. The weightof an atom is due to the protons and neutrons (the electrons are negligible inweight), thus the atomic weight is virtually equal to the sum of the protons and theneutrons.

16.2.3.3 Electrons

The electron orbits define the size or volume occupied by the atom. Theelectrons travel in orbits which are many times the diameter of the nucleus andhence the space occupied by an atom is virtually empty! The electrical propertiesof the atom are determined by how tightly the electrons are bound by electricalattraction to the nucleus.

16.3 PERIODIC TABLE


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