Pre ATPL Maths and Physics Revision Course-1
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Maths & Physics Review for Pilots
Knowledge of mathematics and physics will play a strong part in the study towards your ground examinations for the award of any pilot’s licence. The aim of this document is to provide a revision package for those who feel that their skills in these fields of study are in need of some refreshment or refinement. The topics covered are used in a number of the subjects within the JAR syllabus and this document will endeavour to use examples and units of measurement that are relevant to the aviation environment. By no means do you have to be a mathematician or a physicist in order to fly a plane, but the knowledge of these subjects will stand you in good stead for passing the theoretical exams in the run up to your license issue. Many pilot selection processes for training schools and companies may issue you with a relatively straight forward maths test so if you can’t pass it within a specified time limit then you may find yourself going no further in the selection process. I highly recommend nearer the time that you not only use this document to brush up on the core mathematic skills but also to revise ‘speed maths’ using just a pencil and paper.
Multiplication Factors The following multiplication factors are used with units of measurement and help to properly refer to quantities. For example, instead of saying ‘1,000 metres’, a person can convert it to ‘1 kilometre’ to clarify communication. Multiplication Factor
1 000 000 = 106 1 000 = 103 100 = 102 0.01 = 10-2 0.001 = 10-3 0.000 001 = 10-6
mega kilo hecto centi milli micro
M k h c m u
Multiple Arithmetic Process If a calculation requires the use of several of the basic arithmetic processes, the rules for the order in which those processes must be completed is as follows:
1. 2. 3. 4.
Take out any brackets Complete any multiplications Carry out any divisions Add values together and subtract values from one another Example (6 x 2 - 2) ÷ 5 + 5 x 6 - 3 = (12 - 2) ÷ 5 + 5 x 6 - 3 = 10 ÷ 5 + 30 - 3 = 2 + 30 - 3 = 32 - 3 = 57 It is also worth remembering that, when using fractions, division by a fraction is the same process as multiplication by the reciprocal (inverse) of that fraction. Example 1/7 ÷ 3/7 = 1/7 x 7/3 = (1 x 7)/(7 x 3) = 7/21 = 1/3 To convert a fraction to a decimal, simply divide the top number (numerator) by the bottom number (denominator)4/7 = 4 ÷ 7 = 3.428 Ratios When two quantities are compared, the result of dividing one by the other is called the ratio of the first quantity to the second. Example The ratio of 8 to 4 = 4/2 or 2/1 This can be expressed in words as: The ratio of 8 to 4 is the same as the ratio of 2 to 1 Or symbolically: 8:42:1 When the ratio between two quantities is known it can be used to calculate the particular value of one quantity corresponding to a given value of the other quantity. Example:
The specific gravity (SG) of a fuel is the ratio of the mass of a given volume of the fuel to the mass of the same volume of water. The SG depends upon the composition of the fuel, but a commonly used value for aviation fuel is 0.72. We know that one imperial gallon of water has a mass of 10 lbs; therefore, if we have 400 imperial gallons of this fuel we can calculate the mass of the fuel as follows: 400 imperial gallons of water has a mass of 4000 lbs Then if the mass of the 400 imperial gallons of the fuel is ‘m’ lbs: 0.73 = m 1 5000 Therefore,
m = 0.73 x 5000 = 3650 lbs
Conversions Conversion of units refers to conversion factors between different units of measurement for the same quantity. The process of conversion depends on the specific situation and the intended purpose. Conversion between units in the metric (SI) system can be discerned by their prefixes (for example, 1 kilogram = 1000 grams, 1 milligram = 0.001 grams). Units conversion by factor-label Many, if not most, parameters and measurements in the physical sciences and engineering are expressed as a numerical quantity and a corresponding dimensional unit; for example: 1000 kg/m³, 100 kPa/bar, 50 miles per hour, 1000 Btu/lb. Converting from one dimensional unit to another is often somewhat complex and being able to perform such conversions is an important skill to acquire. The factorlabel method, also known as the unit-factor method or dimensional analysis is an organized way to perform conversions. The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors as shown below: 10 mile 1609 meter 1 hour meter -- ---- × ---- ----- × ---- ------ = 4.47 -----1 hour 1 mile 3600 second second
It can be seen that each conversion factor is equivalent to the value of one. For example, starting with 1 mile = 1609 meters and dividing both sides of the equation by 1 mile yields 1 mile / 1 mile = 1609 meters / 1 mile, which when simplified yields 1 = 1609 meters / 1 mile. So, when the units, mile and hour, are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.47 meters per second. Arithmetic The ability to do quick arithmetic without halting or stumbling is important and must be practiced. The following exercises are to be attempted during a timed session and then tested for accuracy. Below are some common question types for selection processes throughout the UK so it’s important that you can become quick at calculating the answers with nothing more than a pencil, calculator and piece of paper. Set your timer for ten minutes and do the following problems: 1. The power generated by an engine is reduced by 8 units for every 1,000 feet above sea level. Given that the power 475 units at sea level, what will the power be at 4000 feet? 2. An aircraft uses 4,900 lbs of fuel per hour. How much fuel will be consumed in 23 minutes? 3. You are consuming 4,000 lbs/hr of fuel during a flight. 1/6 hour before landing, you still have 17,000 lbs. of fuel. How many pounds of fuel will you have upon landing? 4. If you are travelling at 215 kts, how far will you travel in 25 minutes? 5. An oil gauge must not be allowed to fall below 10 pts. If, during a flight, your oil gauge reads 35 pts . What is the lowest acceptable amount if you need to continue for another 2¼ hours if you take another reading after 0.5 hour? 6. You have access to a freight area of 20 square yards. How many 2 ft x 5 ft boxes can be placed on the floor? 7. What is the fuel consumption of a 4 engines aircraft in lb per hour, if each engine’s average consumption is 18.3 lb/min?
8. If you are travelling at 80 kts, how long will it take to travel 45.5 nautical miles?
Answers 1. 443 units 2. 1878 lbs. 3. 16,333 lbs. 4. 89.6 5. 30.5 pts. 6. 18 7. 4392 lbs/hr 8. 34 minutes
Percentages In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or 0.45. Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. Percentage is a very good technique to see how much a task has been completed. e.g a task takes 20 hours to be done completely. If the task is 70% done it means (70%=(x/20)*100 so x=70*20/100=14 hours) 14 hours work is completed and 6 hours work left. Examples: What is 200% of 30? Answer: 200% × 30 = (200 / 100) × 30 = 60. What is 13% of 98? Answer: 13% × 98 = (13 / 100) × 98 = 12.74. 60% of all university students are male. There are 2400 male students. How many students are in the university?
Answer: 2400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000. There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village? Answer: 75 = X% × 300 = (X / 100) × 300, so X = (75 / 300) × 100 = 25, and therefore X% = 25%. The number of students at the university increased to 4620, compared to last year's 4125, an absolute increase of 495 students. What is the percentual increase? Answer: 495 = X% × 4125 = (X / 100) × 4125, so X = (495 / 4125) × 100 = 12, and therefore X% = 12%. The required level runway length for an aircraft to take-off safely under specified aircraft conditions is 5000ft. However, the runway in use actually has an upslope of 2º and this increases the required runway length by 10%. Calculate the runway length required under these circumstances. Required runway length = 5000 + 10% of 5000 = 5000 + (5000 x 0.1) = 5000 + 500 = 5500 ft The total fuel capacity of an aircraft is 2000 litres. If 25% of the fuel used in a flight must be kept in reserve, calculate the maximum amount of fuel available for a flight, assuming that the reserve fuel is unused. The most common error is to say that the reserve fuel is 25% of the total fuel (2000 litres). This gives the reserve fuel as 0.25 x 2000 = 500 litres, leaving 2000 – 500 = 1500 litres as the maximum flight fuel available. This is incorrect! The correct starting point for the calculation is: Total fuel = Flight fuel + Reserve fuel = Flight fuel + 25% of Flight fuel 2000 = Flight fuel + (0.25 x Flight fuel) = 1.25 x Flight fuel 2000/1.25 = Flight fuel Maximum Available Flight fuel = 1600 litres Graphs Graphs are pictorial solutions to relationships and a simple graph has two axes, commonly referred to as the x-axis and the y-axis, that are offset by 90º degrees to one another. Any two sets of values can be plotted (one on each axis) as they vary and a line drawn joining the sets will form the graph of the relationship between them. For example, in meteorology air temperature normally decreases as altitude increases and the result of measuring the temperature at a selection of different altitudes can be plotted on a graph. The result is known as a temperature/height diagram. If height is plotted along the vertical axis and temperature along the horizontal axis the graph might look something like this:
Having drawn the graph it is now possible to find the temperature at any height represented on the height axis (vertical axis) of the graph. For example, the temperature at a height of 25’000ft is about -48ºC. Alternatively, it is possible to find the height at which the temperature is a particular value. For example, the height at which the temperature is -26ºC is about 17’000ft. Variation If we take an expression such as A = B / C then it is possible to state how one part of the expression will vary with a change in the value of one of the other variables in the expression. If b increases then a will also increase and if b decreases then a will also decrease (assuming c to stay constant). From the above deductions we say that a varies directly with b. On the other hand if c increases then a will decrease and if c decreases then a will increase (assuming b to be constant). From this we can say that a varies inversely or indirectly with c.
If two quantities are known to vary directly or inversely we are able to calculate the change in one quantity corresponding to a known change of the other quantity, if the original values of the two quantities are known. When two quantities vary directly the result of dividing one into the other is a constant. If a varies directly as b then: Old a = New a = constant Old b New b This can be expressed in the format: New a = Old a x New b Old b When two quantities vary inversely the result of multiplying them together is a constant. If a varies inversely as c then: Old a x Old c = New a x New c = constant This is expressed in the format: New a = Old a x Old c New c Example: Given the expression A = 27.3 x B x C and that A is 12 when B is 7. QxM Calculate the value of A when B decreases to 5, if the other terms remain constant. From the expression, A varies directly as B Therefore, New A = Old A x New B = 12 x 5/7 = 60/7 = 8.5714 (A and B both decrease) Old B Using the same expression and given that M is 2 when A is 15, calculate the value of M when A increases to 20, if the other terms remain constant. From the expression, A varies inversely as M Therefore, New M = Old M x Old P = 2 x 15/20 = 1.5 (M decreases when A increases) New P
Pythagorus Theorem states that the square on the hypotenuse of a right-angled triangle (the longest side opposite the right angle) is equal to the sum of the squares on the other two sides.
b A Angle C = 90º c is the hypotenuse
c2 = a2 + b2
Trigonometric Functions The three sides of a right-angled triangle are given names relative to the included angle under examination. The adjacent side is the side along the side of the angle, the opposite side is the hypotenuse which has already been defined. The diagram below illustrates these concepts:
Hypotenuse Opposite Side
The common trigonometric functions are as follows:
Sine(Sin) = Opposite side Hypotenuse
Cosine(Cos) = Adjacent side Hypotenuse
Tangent(Tan) = Opposite side Adjacent side
Cosecant(Cosec) = 1/sine = Hypotenuse Opposite side
Secant(Sec) = 1/cosine = Hypotenuse Adjacent side
Cotangent(Cot) = 1/tangent = Adjacent side Opposite side
The following relationships can also prove useful in calculations:
SinA = Cos(90-A) CosA = Sin(90-A) TanA = Cot(90-A)
(Where A represents an angle in a triangle) Simple Trigonometry “SOH-CAH-TOA” Soh-Cah-Toa can be used to determine the derivation of the equation required to calculate an answer to a simple algebra question (specifically referring to right-angled triangles). As part of your study towards a Pilot’s License you will certainly encounter simple algebra type questions in such topics as Meteorology (wind components), Principles of Flight (aerofoils and airflow) and Navigation (track lines and headings).
SOH Soh stands for Sine of Angle, Opposite, Hypotenuse. The Hypotenuse is the side that is not perpendicular to any other side. The Opposite is the side that is opposite the angle. The Sine of angle is just that - the Sine of the Angle. You need 2 of these variables to calculate the third but what you really need is the equation that links the 3 of them…
The symbol in the triangle is the symbol for an angle. Just move the equation round to find which parts you need i.e. 1.
Sine x Hyp. = Opp.
2. Opp. / Sine = Hyp. You cannot use the right angle as the angle in the equation. If you need to find out the Sine of the angle and not the angle itself then you must use a function called the inverse of Sine (normally written as Sine to the power of -1 on a calculator.
CAH Like Soh, Cah is also an acronym. Cosine of Angle, Adjacent, Hypotenuse. This works the same as Soh but just using the side adjacent to the angle instead of opposite to it and you should find the Cosine of the angle instead of the Sine. There is also an inverse of Cosine. The method of Cah is the same as Soh, just rearrange the part to find what you need.
TOA Toa stands for Tangent (written as Tan) of Angle, Opposite, Adjacent. If you haven't spotted it yet, the middle letter of each acronym so far has been the numerator on the right and the and last letter the denominator. Remember this and SOH-CAH-TOA and you'll develop an excellent understanding of Trigonometry in no time. Tan also has an inverse.
SI Units The SI is a metric system used in science providing a complete metric system for units of measurement and is based on the following fundamental units. Not all units have been included due to irrelevance to the JAR syllabus:
Length Mass Time Temperature Electric current
metre (m) kilogram (kg) second (s) Kelvin (k) ampere (A)
All other SI units can be derived in terms of one or more of the fundamental units. The following derived units are commonly used:
Frequency Energy Force Pressure Power Electric Charge Potential Difference Capacitance
hertz (Hz) joule (J) Newton (N) pascal (Pa) watt (W) coulomb (C) volt (V) farad (F)
Vector Mechanics In physics, vectors are used to represent physical quantities which have both a magnitude and direction, such as force, in contrast to scalar quantities, which have no direction. For example, forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous. For example, if you know that two people are pulling on the same rope with known magnitudes of force but you do not know which direction either person is pulling, it is
impossible to determine what the acceleration of the rope will be. The two people could be pulling against each other as in tug of war or the two people could be pulling in the same direction. In this simple one-dimensional example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other. Associating forces with vectors avoids such problems.
Addition and subtraction Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is
The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below:
This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, it will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is
Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the end points of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below:
Resolution of Vectors Vectors can also be ‘resolved’ into components by reversing the process of addition previously explained. Once again it is common to carry out the process diagrammatically and it’s common to resolve a vector into only two components that are usually at right angles to each other. For example the movement of air over the aircraft wing produces pressure changes that create an aerodynamic force (a vector) called the ‘Total Reaction’. Lift is defined as the component of the Total Reaction that acts perpendicular to the relative airflow over the wing and drag as the component of the Total Reaction that acts in the same direction as the relative airflow. The Total Reaction can be resolved into its components of lift and drag diagrammatically and the principle is illustrated below.
The Total Reaction has been resolved into two ‘components’ of Lift and Drag. It can be seen that the sum of Lift and Drag, using the vector addition method is the Total Reaction. Common Physical Properties Velocity The velocity of a body is defined as its rate of change of position with respect to time with the direction of the motion being stated. If the body is travelling in a straight line it is in linear motion, and if it covers equal distances in equal successive time intervals it is in what is known as uniform linear motion. For uniform velocity, where s is the distance covered in time t, the velocity is given by: v=s t The SI unit of velocity is ms-1 (metres per second) although in aviation this would usually be expressed using the knot (one nautical mile per hour). Acceleration The acceleration of a body is its rate of change of velocity with respect to time. Any change in either speed or direction of motion involves an acceleration. When the velocity of a body changes by equal amounts in equal intervals of time it is said to have a uniform acceleration. If the initial velocity u of a body in linear motion changes uniformly in time t to velocity b, its acceleration a is given by: a = (v – u) t The SI unit of acceleration is ms-2 (metres per second squared). Mass The mass of a body may be defined as the quantity of matter within the body. The SI unit of mass is kg (kilogram). Density The density of a substance is the mass per unit volume of the substance. The SI unit of density is kgm-3 (kilogram per cubic metre).
Pressure Pressure is the force per unit area acting on the surface of a body. The SI unit of pressure is the pascal and in terms of the fundamental SI units, the unit of pressure is kgm-1s-2 (kilogram per metre per second squared). Force A force is a quantity which when acting on a free moving body produces an acceleration in the motion of that body and is proportional to the rate of change of momentum of the body. Force (F) is expressed in terms of the mass (m) and acceleration (a) of a body by the formula: F = ma The SI unit of force is the Newton and the common derived unit of force is kgms-2 (kilogram metres per second squared). One Newton is the force that gives an acceleration of one metre per second squared to a mass of one kilogram. Weight The weight of a body is a measure of the force of acceleration that is exerted on its mass, more commonly on earth- gravity acting on the mass. The SI unit of weight is the same as that of force (the Newton) and the gravitational constant generally applied is the acceleration due to gravity (g) is equal to 9.81 ms-2. Inertia Inertia is the tendency of a body to maintain its state of rest or uniform motion in a straight line. The application of an outside force is necessary to overcome the inertia of a body either at rest or in a state of uniform motion. Work Is the amount of energy transferred by a force acting through a distance. The unit of work is the joule and in terms of the fundamental SI units, the unit of work is kgm2s2 (kilogram metres squared per second squared). One joule is the work done when a force of 1 Newton moves a mass of 1 kilogram by 1 metre in the direction of the force. Momentum The momentum of a body is defined as the product of its mass and its velocity. Momentum is a vector quantity and a change in speed or direction constitutes a
change in the momentum. The SI unit of momentum is kgms-1 (kilogram metre per second). Power Power is defined as the rate of doing work measured in units of work per unit time. The SI unit of power is the watt and but the more commonly derived unit of power is kgm2 s-3 (kilogram metres squared per second cubed). Friction Friction is the force resisting the relative lateral (tangential) motion of solid surfaces, fluid layers, or material elements in contact. Friction is not a fundamental force, as it is derived from electromagnetic force between charged particles, including electrons, protons, atoms, and molecules, and so cannot be calculated from first principles, but instead must be found
Coefficient of friction The coefficient of friction (COF), also known as a frictional coefficient or friction coefficient and symbolized by the Greek letter µ, is a dimensionless scalar value which describes the ratio of the force of friction between two bodies and the force pressing them together. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than one – under good conditions, a tire on concrete may have a coefficient of friction of 1.7 Atmospheric Pressure Atmospheric pressure is defined as the force per unit area exerted against a surface by the weight of air above that surface at any given point in the Earth's atmosphere. In
most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point. Low pressure areas have less atmospheric mass above their location, whereas high pressure areas have more atmospheric mass above their location. Similarly, as elevation increases there is less overlying atmospheric mass, so that pressure decreases with increasing elevation. The standard atmosphere (symbol: atm) is a unit of pressure and is defined as being equal to 101,325 Pa or 101.325 kPa. The following units are equivalent, but only to the number of decimal places displayed: 760 mmHg (torr), 29.92 inHg, 14.696 PSI, 1013.25 millibars. Mean sea level pressure (MSLP) is the pressure at sea level or (when measured at a given elevation on land) the station pressure reduced to sea level assuming an isothermal layer at the station temperature. This is the pressure normally given in weather reports on radio, television, and newspapers or on the Internet. When barometers in the home are set to match the local weather reports, they measure pressure reduced to sea level, not the actual local atmospheric pressure. The altimeter setting in aviation, set either QNH or QFE, is another atmospheric pressure reduced to sea level, but the method of making this reduction differs slightly. QNH The barometric altimeter setting which will cause the altimeter to read airfield elevation when on the airfield. In ISA temperature conditions the altimeter will read altitude above mean sea level in the vicinity of the airfield QFE The barometric altimeter setting which will cause an altimeter to read zero when at the reference datum of a particular airfield (generally a runway threshold). In ISA temperature conditions the altimeter will read height above the datum in the vicinity of the airfield. QFE and QNH are arbitrary Q codes rather than abbreviations, but the mnemonics "Nautical Height" (for QNH) and "Field Elevation" (for QFE) are often used by pilots to distinguish them. Gas Laws Boyles Law Boyle's Law shows that, at constant temperature, the product of an ideal gas's pressure and volume is always constant. It can be determined experimentally using a pressure gauge and a variable volume container. It can also be found logically; if a container
with a fixed amount of molecules inside it is reduced in volume, more molecules will hit the sides of the container per unit time causing a greater pressure. As a mathematical equation, Boyle's law is: P1V1 = P2V2
V1 is the original volume V2 is the new volume P1 is original pressure P2 is the new pressure
Charles Law Charles's Law states that the volume occupied by any sample of gas at a constant pressure is directly proportional to the absolute temperature. V / T =constant
V is the volume T is the absolute temperature (measured in Kelvin)
Charles's Law can be rearranged into two other useful equations. V1 / T1 = V2 / T2
V1 is the initial volume T1 is the initial temperature V2 is the final volume T2 is the final temperature
V2 = V1 (T2 / T1)
V2 is the final volume T2 is the final temperature V1 is the initial volume T1 is the initial temperature
Charles's Law only works when the pressure is constant. Moments Torque, also called moment or moment of force, is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.
In more basic terms, torque measures how hard something is rotated. For example, imagine a wrench or spanner trying to twist a nut or bolt. The amount of "twist" (torque) depends on how long the wrench is, how hard you push on it, and how well you are pushing it in the correct direction. The terminology for this concept is not straightforward: In physics, it is usually called "torque", and in mechanical engineering, it is called "moment". The magnitude of torque depends on three quantities: First, the force applied; second, the length of the lever arm connecting the axis to the point of force application; and third, the angle between the two. The principle of moments is illustrated below. AC is a plank resting on a pivot B. A force of 75 N acts vertically upwards through A and a force of 50 N through C. 75 N 50 N B A
If AB = 10 cm and BC = 7 cm then taking moments about B will ( if we assume the convention that a clockwise rotation is positive and an anticlockwise negative) give: Total moment = (75 x 0.1) – (50 x 0.7) = 7.5 – 3.5 = 4 Nm Centimetres having been converted into Metres Because the sign of the total moment is positive, the two forces together would rotate the plank in a clockwise direction. The magnitude of the moment is a measure of the rate of rotation of the plank (the greater the magnitude then the greater the rate of rotation). If the total moment about a point is zero the system is said to be in ‘equilibrium’ with respect to moments. The whole system is in equilibrium if the vector sum of the forces involved is zero. Couples A Couple is a system of forces with a resultant (a.k.a. net, or sum) moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. The resultant moment of a couple is called a torque. This is not to be confused the term torque as it is used in physics, where it is merely a synonym of moment. Instead,
torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point. Simple couple The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple". The forces have a turning effect or moment called a torque about an axis which is normal to the plane of the forces. The SI unit for the torque of the couple is Newton Metre. Centre of Gravity The centre of mass of a system of particles is the point at which the system's whole mass can be considered to be concentrated for the purpose of calculations. The centre of mass is a function only of the positions and masses of the particles that compose the system. In the case of a rigid body, the position of its centre of mass is fixed in relation to the object (but not necessarily in contact with it). The centre of mass is often called the centre of gravity but this is only true in a system where the gravitational forces are uniform. Mass & Balance Some common words used within the subject area when calculating mass and balance figures prior to loading an aircraft. Ballast Ballast is removable or permanently installed weight in an aircraft used to bring the centre of gravity into the allowable range. Centre-of-gravity (CG) limits CG limits are specified longitudinal (forward and aft) and/or lateral (left and right) limits within which the aircraft's centre of gravity must be located during flight. The CG limits are indicated in the airplane flight manual. The area between the limits is called the CG range of the aircraft. Weight and balance When the weight of the aircraft is at or below the allowable limit(s) for its configuration (parked, ground movement, take-off, landing, etc.) and its centre of gravity is within the allowable range, and both will remain so for the duration of the flight, the aircraft is said to be within weight and balance. Different maximum weights may be defined for different situations; for
example, large aircraft may have maximum landing weights that are lower than maximum take-off weights (because some weight is expected to lost as fuel is burned during the flight). The centre-of-gravity may change over the duration of the flight as the aircraft's weight changes due to fuel burn. Reference datum The reference datum is a reference plane that allows accurate, and uniform, measurements to any point on the aircraft. The location of the reference datum is established by the manufacturer and is defined in the aircraft flight manual. The horizontal reference datum is an imaginary vertical plane or point, arbitrarily fixed somewhere along the longitudinal axis of the aircraft, from which all horizontal distances are measured for weight and balance purposes. There is no fixed rule for its location, and it may be located forward of the nose of the aircraft. For helicopters, it may be located at the rotor mast, the nose of the helicopter, or even at a point in space ahead of the helicopter. While the horizontal reference datum can be anywhere the manufacturer chooses, most small training helicopters have the horizontal reference datum 100 inches forward of the main rotor shaft centreline. This is to keep all the computed values positive. The lateral reference datum, is usually located at the centre of the helicopter. Arm The arm is the horizontal distance from the zero point of the datum to any component of the aircraft or to any object located within the aircraft. Other terms used interchangeably with arm are station and centroid (used on large transport category aircraft). Moment The moment is a measure of force that results from an object’s weight acting through an arc that is centred on the zero point of the reference datum distance. Moment is also referred to as the tendency of an object to rotate or pivot about a point (the zero point of the datum, in this case). The further an object is from this point, the greater the force it exerts. Moment is calculated by multiplying the weight of an object by its arm. Mean Aerodynamic Chord (MAC) A specific chord line of a tapered wing. At the mean aerodynamic chord, the centre of pressure has the same aerodynamic force, position, and area as it does on the rest of the wing. The MAC represents the width of an equivalent rectangular wing in given conditions. On some aircraft, the centre of gravity is expressed as a percentage of the length of the MAC. In order to make such a calculation, the position of the leading edge of the MAC must be known ahead
of time. This position is defined as a distance from the reference datum and is found in the aircraft's flight manual and also on the aircraft's type certificate data sheet. If a general MAC is not given but a LeMAC (leading edge mean aerodynamic chord) and a TeMAC (trailing edge mean aerodynamic chord)are given (both of which would be referenced as an arm measured out from the datum line) then your MAC can be found by finding the difference between your LeMAC and your TeMAC. Calculation Center of gravity is calculated as follows: 1. 2. 3. 4.
Determine the weights and arms of all mass within the aircraft. Multiply weights by arms for all mass to calculate moments. Add the moments of all mass together. Divide the total moment by the total weight of the aircraft to give an overall arm.
The arm that results from this calculation must be within the arm limits for the center of gravity that are dictated by the manufacturer. If it is not, weight in the aircraft must be removed, added (rarely), or redistributed until the center of gravity falls within the prescribed limits. In larger aircraft, weight and balance is often expressed as a percentage of mean aerodynamic chord, or MAC. For example, assume that by using the calculation method above, the center of gravity (CG) was found to be 76 inches aft of the aircraft's datum and the leading edge of the MAC is 62 inches aft of the datum. Therefore, the CG lies 14 inches aft of the leading edge of the MAC. If the MAC is 80 inches in length, the percentage of MAC is found by calculating what percentage 14 is of 80. In this case, one could say that the CG is 17.5% of MAC. If the allowable limits were 15% to 35%, the aircraft would be properly loaded. Example Given: Weight (lb) Arm (in) Moment (lb-in) Empty weight
Pilot and passengers
Fuel (30 gallons @ 6 lb/gal) 180.0
To find the centre of gravity, we divide the total moment of mass by the total mass of the aircraft: 193,193 ÷ 2,055 = 94.01 inches behind the datum plane.