Dimensions and Units Lecture Notes

Dimensions and Units Goals of these notes After studying these notes you should be able to, 1. understand and describe the difference between dimensions and units, 2. convert a physical quantity from one set of units to another, 3. recognize the difference between force and mass even when similar units – lbm and lbf – are used for both quantities in the engineering system units, 4. convert temperatures and temperature differences from relative to absolute units, 5. carry units in calculations and cancel them using algebra as a check on units, and 6. be able to use the common units for mass, length, time, force, pressure, energy and power in both the SI units and engineering units. Introduction to dimensions and units Measured physical properties have a basic dimension in which they are measured. There may be many units that are used to measure this dimension. This is best shown by example. The thickness of an object has the dimension of length. Length can be measured in a wide range of units including inches, feet, yards, meters, kilometers, micrometers, Angstrom units, furlongs, fathoms, light-years and many more. The thickness of an object cannot be measured in kilograms, however. That is because the kilogram is a unit used to measure quantities that have the fundamental dimension of mass. The choice of the units to use in a particular measurement is a matter of convention or convenience. Whatever units you use, however, must correspond to the correct dimension for the physical quantity. Systems of units usually start by making arbitrary definitions of a unit for fundamental dimensions. Typically these fundamental dimensions are mass, length, time, electric charge and temperature. Once these units are selected for the fundamental dimensions the units for other physical quantities can be determined from the physical relations among quantities having the fundamental units. For example velocity is found as distance divided by time. Thus the dimensions of velocity must be length/time. Similarly the dimensions of acceleration, found as velocity divided by time, must be length/time2, and the dimensions of force can be found from Newton's second law: force equals mass times acceleration. This gives the dimensions of force as the dimensions of mass times the dimensions of acceleration or (mass) times (length) divided by (time)2. The symbols M, L, and T are often used to represent dimensions of mass, length, and time, respectively. However, these symbols are not used in these notes. We can continue in this fashion. Pressure is force per unit area; this must have dimensions of (force) / (length)2 = (mass) (length) / [(time)2(length)2] = (mass) / [(time)2(length)]. Work is the product of force times distance. Thus the dimensions of work must be the dimensions of force times the dimensions of distance. This means that the dimensions of work are (mass)(length)2/(time)2. Because work is a form of energy we should expect that any kind of energy should have the same dimensions. We can check this by recalling the formula for kinetic energy: mV2/2. This has the dimensions of mass times the dimensions of velocity squared. But, this is just (mass) times (length/time)2, which is the result that we just found for work. Similarly, potential energy, mgz, has dimensions of mass times dimensions of acceleration times dimensions of length. This gives (mass) times [(length)/(time)2] times (length). Again the same result: the dimensions of energy are (mass) (length)2 /(time)2. Finally, power which is energy divided by time must have dimensions of (mass) (length)2 /(time)3. SI units * In the SI system of units the first three fundamental dimensions are mass, length and time. The units used to measure these dimensions (and their abbreviations) are defined to be the kilogram (kg), the meter (m) and the second (s). With these definitions the units for velocity and acceleration become meters per second (m/s) and meters per second per second or meters per second2 (m/s2), respectively. The kilogram is defined as the mass of a reference material located at the International Bureau of Weights and Measures. The meter is defined as the length of a path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second. (This is equivalent to defining the speed of light as 299,792,458 m/s.) The second is defined in terms of the wavelength of a certain emission from a cesium atom and the kelvin (temperature) is defined as 1/273.16 of the triple point temperature of water. † The ampere is defined as the basic unit of electrical current. As mentioned above the force units must be the same as the mass units times the acceleration units. Thus the SI units of force are kg•m/s2. These units are given a special name, the newton (N). Note that we use a capital N when we spell the last name of Sir Isaac Newton, but a lower case n when we spell the name of the SI force unit, the newton. However, we use an upper case N when we abbreviate one newton as 1 N. This is the common practice in modern unit naming conventions. This definition of the newton can be written as a unit-conversion factor equation: 1 N = 1 kg•m•s-2 Similarly the units for pressure, energy and power, which can be written in terms of fundamental units, are also given separate names for convenience. Table 1 gives a summary of the various units in the SI system of units. Physical quantity mass length time force pressure energy power Unit name kilogram meter second newton pascal joule watt Table 1 Abbreviation kg m s N Pa J W Definition fundamental unit fundamental unit fundamental unit 1 N = 1 kg•m•s-2 1 Pa = 1 N •m-2 = 1 kg•m-1•s-2 1 J = 1 N•m = 1 kg•m2•s-2 1 W = 1 J•s-1 = 1 N•m•s-1 = 1 kg•m2•s-3 Prefixes The wide scale of units in engineering applications is handled by using prefixes that account for multiplicative factors on the SI units. Thus a kilopascal (abbreviation kPa) means 1000 pascals. The following prefixes, and their associated factors and abbreviations are commonly used with SI units. Prefix Factor Abbreviation Prefix Factor Abbreviation deci 10-1 d deka 10 da Table 2 – Prefixes for SI units centi milli micro nano 10-2 10-3 10-6 10-9 c m n µ hecto kilo mega giga 102 103 106 109 h k M G pico 10-12 p tera 1012 T femto 10-15 f peta 1015 P atto 10-18 a exa 1018 E Note that in areas and volumes we speak of square kilometers or cubic millimeters. We apply the prefix to the length measurement. So parallelepiped that has sides of 7 mm, 12 mm, and 25 mm has a volume of 2,100 mm3. This is a volume of 2.1x10-6 m3. The kilogram is an unusual SI unit. Although it is a basic unit, it has the kilo prefix, because it was originally defined as 1000 grams. For purposes of naming mass units, the gram = 10-3 kilograms provides the base name. Thus we can speak of megagrams, micrograms, and the like rather than using a prefix in front of kilograms. Engineering units In the engineering system of units the word pound is used to define both a unit of force and a unit of mass. To distinguish between the two we will speak of a pound-force (lbf) and a pound-mass (lbm). ‡ The pound-mass is defined in terms of the kilogram as a mass unit. One pound mass equals 0.4535924 kilogram. The basic unit of length, the foot, is defined to be exactly 0.3048 meters. The time unit, the second, is the same as in SI units. If we were to define a force unit in the same way as defined in SI units we would create a name for the mass times acceleration units of 1 lbm•ft•s-2. In fact there is a name for this quantity. It is called a poundal; a name found only in obscure physics texts. The usual unit for force, the pound-force, is defined as the force that is generated when one pound mass is weighed in a standard gravitational field § of 32.174 ft•s-2. Thus we define 1 lbf = 32.174 lbm•ft•s-2 NIST publications use the name pound and symbol lb for the pound mass while using the name poundforce and the symbol lbf for the pound force. ‡ § The standard gravitational acceleration is defined to be exactly 9.80665 m/s2. Because this unit conversion factor is based on a standard gravitational acceleration, it is a fixed number, which is always the same regardless of the local gravitational acceleration. The weight of a mass is the product of the mass times the local gravitational acceleration. At a point where the local gravitational acceleration, g = 32.11 ft/ s2, the weight of a 100 pound mass would be given as W = (100 lbm ) ( 32.11 ft/ s2.) = 3211 lbm• ft/ s2 We can convert these unwieldy units to a more conventional one by using the unit conversion factor for pound force. A convenient way to use unit conversion factors is to write them as a ratio equal to unity. Thus we would write: 1 = 32.174 lbm•ft /(lbf•s2) We can multiply anything by 1 and not change the result; so we can multiply our answer of 3211 lbm• ft/ s2, found above, by the unit conversion factor written in this form to obtain W= 3211 lbm ⋅ ft s2 lb f ⋅ s 2 32.174 lbm ⋅ ft = 99.8 lb f Note that we can regard the abbreviations for the units as algebraic symbols to be cancelled. When we do this we wind up with the correct set of units. ** In a similar manner we could compute the kinetic energy of a 2000 pound mass moving at a velocity of 100 ft/s as follows: 2 lb f ⋅ s 2 mV 2 2000 lbm  100 ft  = KE = = 3.11x105 ft ⋅ lb f   2 2  s  32.174 lbm ⋅ ft Again we see that the use of the unit conversion factor and the canceling of the unit abbreviations as algebraic quantities give us the result in the units we desire. Note that an answer of 107 lbm • ft/s2 would have the correct dimensions, but the units are not the conventional ones. The technique outlined above for algebraic cancellation of units is a helpful tool for ensuring that you have correct units when you are working with new equations or new systems of units. The English system of units has no formal name for the energy unit of ft•lbf. There is a commonly used energy unit, the British thermal unit or Btu ††, which is defined in terms of ft•lbf as follows: 1 Btu = 778.169 ft-lbf = 1.055056 kJ In addition to the usual SI unit of watts for power, the engineering system of units sometimes uses the unit of horsepower (hp). The unit conversion factors for energy are: ** An alternative approach for engineering units is to take the pound force as a defined term, equal to 4.448222 N and define the slug as the mass that requires a force of 1 lbf when it is accelerated at 1 ft/s2. This unit is sometimes used in fluid dynamics. What is the conversion factor between slugs and lbm? Another approach is do define the poundal as the force required to accelerate 1 lbm at 1ft/s2. This unit is not in common use for engineering analyses. What is the conversion factor between newtons and poundals? There are actually several different definitions of the Btu that cause differences in the fourth place of the unit conversion factors. This is because the Btu was once defined as the amount of heat required to raise one pound (mass) of water one degree Fahrenheit. Depending on the initial temperature of the water, you will get a slightly different measure of the Btu. The conversion factor provided here is for the international table (IT) Btu, which is a standard definition used for thermodynamic tables. This is defined to be exactly 4.184 calories. †† 1 hp = 550 ft•lbf / s = 2544.433 Btu / hr = 0.7456999 kW 1 kW = 3412.14 Btu / hr Problem solving It is useful to use the units for each quantity that is being used in calculations to ensure that the answer is correct. For problems with SI units, a simple rule of thumb, that almost always works, is to use kJ for energy, kPa for pressure and m3 for volume. This will give consistent answers. To do this, convert all data given in other units (e.g., MJ for energy, MPa for pressure or L for volume) into the three units listed above. The one exception to be aware of is the computation of kinetic energy per unit mass as V2/2. If the velocity is in m/s, V2/2 will have units of m2/s2 which is equivalent to J/kg. Your answer must be divided by (1000 J/kJ) to get an answer in kJ/kg. Typical engineering units have energy terms in Btu, pressures in lbf/in2 (psia), and volumes in ft3. Work terms that are computed as ∫PdV will have units of lbf∙ft3/in2 = psia∙ft3. Before adding such a result to an energy term in Btu, multiply the result by the conversion factor (0.18505 Btu/psia∙ft3) to have a result in Btu. When the kinetic energy per unit mass is computed from a velocity in ft/s, the result (in ft2/s2) should be multiplied by the conversion factor (3.99411x10-5 Btu∙s2/(lbm∙ft2)) to get a result in Btu/lbm. Practical energy use In terms of a practical feeling for these quantities you might want to consult your utility bills. Natural gas is billed in units of therms; 1 therm is 105 Btu. Electricity is billed in kW-hr. One kWhr is 3.6 MJ. Thus one Btu is a small amount of energy and a joule is about 1/1000th of a Btu. Around January, 2007, typical prices for homes in the San Fernando Valley were $8.8/GJ for natural gas and $29/GJ for electricity. The energy content of gasoline varies depending on the composition. At a price of $3/gallon the energy cost of a typical gasoline is $26/GJ. (Without California gasoline and sales taxes of $0.585 per gallon the cost would be $21/GJ.) The annual world energy use is about 450 exajoules or 450x1018 J. The US use is about 25% of this total.