Appendix A Units and Dimensions It is essential to distinguish between units and dimensions. Broadly speaking, physical parameters have the separate properties of size and dimension. A unit is a, more or less arbitrarily defined, amount or quantity in terms of which a parameter is defined. A dimension represents the definition of an inherent property, independent of the system of units in which it is expressed. For example, the dimension mass expresses the amount of material of which a body is constructed, the distance between the wing tips (wingspan) of an aeroplane has the dimension length. The mass of a body can be expressed in kilograms or in pounds, the wingspan can be expressed in metres or in feet. Many systems of units exist, each of which with their own advantages and drawbacks. Throughout this book the internationally accepted dynamical system SI is used, except in a few places as specially noted. The Imperial set of units still plays an important roll in aviation, in particular in the United States. Fundamental dimensions and units Dimensions can be written in symbolic form by placing them between square brackets. There are four fundamental units in terms of which the dimensions of all other physical quantities may be expressed. Purely mechanical rela- tionships are derived in terms of mass [M], length [L], and time [T]; thermo- dynamical relationships contain the temperature [θ] as well. A fundamental equation governing dynamical systems is derived from Newton’s second law of motion. This states that an external force F acting on a body is proportional to the product of its mass m and the acceleration a produced by the force: F = kF ma. The constant of proportionality kF is de- 511 512 A Units and Dimensions Table A.1 Dimensions and SI units used for dynamical systems. Quantity Dimension Unit name Symbol Explanation length [L] metre m fundamental unit mass [M] kilogram kg fundamental unit time [T] second s fundamental unit area [L2]– m2 length×length volume [L3]– m3 area×length velocity [LT−1]– ms−1 length/time acceleration [LT−2]– ms−2 velocity/time moment of inertia [ML2]– kgm2 mass×area density [ML−3]– kgm−3 mass/volume mass flow rate [MT−1]– kgs−1 mass/time force [MLT−2]NewtonN,kgms−2 mass×acceleration moment [ML2T−2] – N m force×length pressure, stress [ML−1T−2] Pascal Pa, N m−2 force/area momentum [MLT−1]– Ns,kgms−1 mass×velocity momentum flow [MLT−2]– N,kgms−2 mass×velocity/time work or energy [ML2T−2] Joule J, N m force×length power [ML2T−3]Watt W,Nms−1 work or energy/time angle 1 radian rad length/length angular velocity [T−1]– rads−1 angle/time angular acceleration [T−2]– rads−2 angular velocity/time frequency [T−1] Hertz Hz 1/time termined by the definition of the units of force, mass and acceleration used in the equation. In general, if the system of units is changed, so also is the constant kF . It is useful, of course, to select the units so that the equation becomes F = ma.Inaconsistent system of units, the force, mass, and time are defined so that kF = 1. For this to be true, the unit of force has to be that force which, when acting upon a unit mass, produces a unit acceleration. International System of Units In most parts of the world the Système International d’Unités, commonly abbreviated to SI units, is accepted for most branches of science and engi- neering. The SI system uses the following fundamental units: Flight Physics 513 • Mass: the kilogram (symbol kg) is equivalent to the international stan- dard held in Sèvres near Paris. • Length: the metre (symbol m), preserved in the past as a prototype, is presently defined as the distance (m) travelled by light in a vacuum in 299,792,458−1 seconds. • Time: the second (symbol s) is the fundamental unit of time, defined in terms of the natural periodicity of the radiation of a cesium-133 atom. • Temperature: the unit Kelvin (symbol K) is identical in size with the degree Celsius (symbol ◦C), but it denotes the absolute (or thermody- namical) temperature, measured from the absolute zero. The degree Cel- sius is one hundredth part of the temperature rise involved when pure water is heated from the triple point (273.15 K) to boiling temperature at standard pressure. The temperature in degrees Celsius is therefore T (C) = T (K) − 273.15. Having defined the four fundamental dimensions and their units, all other physical quantities can be established, as in Table A.1. Velocity, for exam- ple, is defined as the distance travelled in unit time. It has the dimension [LT−1] and is measured in metres per second (m s−1, or m/s). The following additional remarks are made in relation to Table A.1: • The SI system defines the Newton (symbol N) as the fundamental unit for force, imparting an acceleration of 1 m s−2 to one kilogram of mass. From Newton’s equation, its dimension is derived as [MLT−2]. By con- trast to some other systems of units, the definition of a newton is com- pletely unrelated to the acceleration due to gravity. Clearly, the SI system forms a consistent system. • The fundamental unit of (gas) pressure or (material) stress is denoted pascal (symbol Pa). The bar is defined as 105 Pa, the millibar1 (mb) amounts to 102 Pa. A frequently used alternative unit of gas pressure is the physical atmosphere (symbol atm), which is equal to the pressure un- der a 760 mm high column of mercury: 1.01325 × 105 Pa. The standard atmosphere is set at an air pressure of 1 atm at sea level. The techni- cal atmosphere (symbol at) is equal to the pressure under a 10 m high column of water, g × 104 Pa. This requires a definition of the accelera- tion due to gravity, which is taken as the value at 45◦ northern latitude: g = 9.80665 m s−2. • The (dimensionless) radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. One radian is equal to 180/π = 57.296◦. 1 The preferred symbol is the hectopascal, hPa. 514 A Units and Dimensions Fractions and multiples Sometimes, the fundamental units defined above are inconveniently large or small for a particular case. In such cases, the quantity can be expressed in terms of some fraction or multiple of the fundamental unit. A prefix attached to a unit makes a new unit. The following prefixes may be used to indicate decimal fractions or multiples of SI units. Fraction Prefix Symbol Multiple Prefix Symbol 10−1 deci d 10 deca da 10−2 centi c 102 hecto h 10−3 milli m 103 kilo k 10−6 micro µ 106 mega M Imperial units Until about 1968, the Imperial (or British Engineering) set of units was in use in some parts of the world, the United Kingdom in particular. It uses the fundamental units foot (symbol ft) for length and pound (symbol lbm) for mass, the unit for time is the second. The corresponding unit for force, the poundal, produces an acceleration of 1 ft s−2 to 1 lbm. The Imperial System is therefore a consistent one. Since the poundal is considered as an unpracti- cally small force, it is often replaced by the pound force (symbol lbf), which is defined as the weight of one pound mass. The pound force is therefore g times as large as the poundal. However, used with 1 pound mass and 1 ft s−2, it does not constitute a consistent set of units. Therefore, the slug has been defined as a mass equal to g times the pound mass, dictating that a standard value is used for the acceleration due to gravity (32.174 ft s−2). The Imperial system uses the Kelvin or the degree Celsius (“centigrade”) as the standard unit of temperature. Although the SI system constitutes the generally accepted international standard, many Imperial units are still in use, especially in the practice of aircraft operation and in the US engineering world. For example, use is still made of the temperature scales Fahrenheit (F) and Rankine (R). The Rankine is an absolute temperature coupled to the Fahrenheit scale and is not to be confused with the former Réaumur temperature unit. The conversion from degrees Fahrenheit to Kelvin is as follows: T (K) = 273.15 + 5/9{T (F) − 32}. The system of units based on the foot, pound, second and rankine is Flight Physics 515 Table A.2 Table for converting British FPSR units into SI units. Quantity Symbol Multiply by to obtain SI units length inch (in) 2.54 × 10−2 m foot (ft = 12 in) 3.048 × 10−1 m mile 1.6093 km nautical mile (nm) 1.8532 km volume cubic ft 2.8317 × 10−2 m3 UK gallon 4.5461 × 10−3 m3 US gallon 3.7854 × 10−3 m3 velocity ft/s 3.048 × 10−1 m/s mile/h 1.609 km/h UK knot = nm/h 1.853 km/h mass slug 1.4594 × 10 kg pound mass (lbm) 4.5359 × 10−1 kg UK ton 1.0165 × 103 kg US short ton 9.0718 × 102 kg force pound force (lbf) 4.4482 N poundal 1.3826 × 10−1 N pressure lbf/in2 (psi) 6.8948 × 103 Pa lbf/ft2 (psf) 4.7880 × 10 Pa temperature Rankine (R) 5/9 K work ft lbf 1.355 Nm energy BTU 1.055 × 103 J specific energy BTU/slug 7.2290 × 10 Nm/kg power slug ft2/s3 1.356 Nm/s horsepower∗ (hp) 7.457 × 102 W viscosity coefficient slug/ft/s 4.788 × 10 kg/m/s kinematic viscosity ft2/s 9.290 × 10−2 m2/s ∗The unit of power in the (former) Technical System of Units is also known as the (metric) horsepower.