The Matheatics 11 Copetency Test More Unit Conversion Exaples In this docuent, we present a few ore exaples of unit conversions, now involving units of easureent both in the SI others which are not in the SI. The purpose of these additional exaples is to illustrate further the general strategy presented in the previous docuent. UNIT CONVERSION FACTORS First, we ust give you a very brief table of soe unit conversion factors for coon physical quantities. This is a very abbreviated list of unit conversion factors. There are any references or hbooks giving uch ore extensive listings. One is the CRC Hbook of Chestry Physics, updated annually by the Checal Rubber Copany. Another is The Metric Guide, published by the Council of Ministers of Education, Canada. You can also find extensive listings of unit conversion factors on the internet. All of the factors given in bold below are exact either by definition or by international convention, according to this second reference. LENGTH WEIGHT/MASS 1 in =.54 c 1 lb = 16 oz 1 = 1 in = 30.48 c 1 lb = 453.5937 g 1 yd = 3 1 ton = 000 lb 1 = 580 1 etric ton = 1000 kg 1 nautical le = 6080 1 carat = 00 g 1 = 1.609344 1 slug = 14.5939 kg AREA ENERGY 1 = 640 acres 1 Btu = 1055.6 J (N = kg /s ) 1 hectare = 10000 = (100 ) 1 calorie = 4.1868 J 1 Joule = 10 7 ergs (g c /s ) VOLUME 1 ip. gal = 4.546090 litres = 4 quarts FORCE 1 ip. quart = 40 fluid ounces 1 Newton = 0.480894 lb 1 bushel = 8 gallons 1 Newton = 10 5 dynes (g c/s ) 1 litre = 1000 c 3 1 U. S. gal (liq) = 3.785306 litres POWER 1 Hogshead = 38.4747 litres 1 horsepower (echanical) 1 Noggin = 14.065 c 3 = 550 lb/s = 745.700 watts NOTES: The following abbreviations are used in the table above: = foot yd = yard = le oz = ounce lb = pound ip gal = iperial gallon There is a fairly fascinating history to the developent of the hundreds of non-si units that have been used in various geographical areas societies in recent centuries (see http://www.unc.edu/~rowlett/units/custo.htl for one account. In particular, as reflected in the table above, the stardization of the gallon, a unit of volue was done differently by the United States the British, resulting in a U. S. gallon which is soewhat saller than the British iperial gallon. Furtherore, the U. S. distinguishes between a gallon for easuring volue of liquids such as wine a gallon for easure volue of dry goods such as grain. David W. Sabo (003) More Unit Conversion Exaples Page 1 of 1
Exaple 1: Convert the length 56.43 to its equivalent in units of etres. We have 1 = 30.48 c 1 = 100 c So, an effective strategy would appear to be c The required teplate is c 56.43 = 56.43 c Filling in the nubers doing the arithetic gives: 56.43 = 56.43 = 30.48 c 1 ( 56.43)( 30.48) 100 1 100 c = 17.199864 Thus, rounding to two decial places, we can state that 56.43 = 17.0 as the final result. Exaple : Highway regulations in a certain jurisdiction state that a truck with two axles ay have a axiu weight of 16000 pounds on the front axle 0000 pounds on the rear axle. Convert these weights to their equivalents in kilogras, rounded to the nearest ten kilogras. Fro our table of unit conversion factors, we have 1 pound = 1 lb = 453.5937 g we know that 1 kg = 1000 g So, an effective strategy would appear to be lb g kg David W. Sabo (003) More Unit Conversion Exaples Page of 1
The teplate for converting the axiu front axle weight is g kg 16000 lb = 16000 lb lb g Filling in the nubers doing the arithetic gives 16000 lb = 16000 lb 453.5937 g 1 lb 1kg 1000 g ( 16000)( 453.5937) = kg = 757.48 kg 1000 The calculation for the axiu rear axle weight goes exactly the sae way: 0000 lb = 0000 lb 453.5937 g 1 lb 1kg 1000 g = 9071.85 kg Thus, rounding to the nearest 10 kg, we can stated that these regulations correspond to a axiu of 760 kg on the front axle, 9070 kg on the rear axle. Exaple 3: According to one internet site, the distance fro Vancouver, B. C. to Hong Kong is 5550 nautical les, as the crow flies. Ignoring the question of whether any crow really could fly that far, deterne what this distance is in units of kiloetres. Fro our list of unit conversion factors, we have that 1 nautical le = 6080 feet 1 foot = 30.48 c we can easily convert centietres to kiloetres. So, an effective strategy here would appear to be naut. c The teplate for this calculation is c 5550 naut. = 5550 naut. naut. c Filling in the nubers copleting the calculation gives David W. Sabo (003) More Unit Conversion Exaples Page 3 of 1
5550 naut. = 5550 naut. 6080 30.48 c 1 1 naut. 1 100 c 1 1000 = ( 5550)( 6080) ( 30.48) ( 100)( 1000) = 1085.171 Since the original length, 5550 nautical les, appears to have just three significant figures, we should probably round our final stated answer to three significant figures as well. So, this distance between Vancouver, B. C. Hong Kong is 10300. Exaple 4: The speed lit on the Coquihalla Highway in British Colubia is 110 /h. What is the equivalent speed in units of /h, rounded down to the nearest ultiple of 5? The easiest approach here is to convert the speed 110 /h to its equivalent in /h, then round the result appropriately. Since the tie units are the sae in both cases, we need only deal with the conversion between kiloetres les. Fro our table of unit conversion factors, we get that 1 = 1.609344 So, it would appear that this proble can be done with the one step strategy The teplate for the calculation is 110 = 110 h h Putting in the nubers doing the calculations gives 110 110 h = 1 h 1.609344 110 = = 68.3508 1.609344 h h Rounding down to the nearest ultiple of 5 give the final answer of 65 /h. Exaple 5: According to the U. S. Departent of Energy, the 003 odel of the Chevrolet Malibu has a highway fuel econoy rating of 9 /gal. Assung this is a reference to the U. S. David W. Sabo (003) More Unit Conversion Exaples Page 4 of 1
gallon, copute the equivalent value of this fuel consuption rate in units of /L. Round the answer to one decial place. Obviously this conversion will involve two parts: U. S. gal (liq) litres Our list of unit conversion factors gives nubers for both of these conversions, so these two lines for an effective strategy for this proble. The teplate for the calculation is gal 9 = 9 gal gal L Here, we ve use the siple gal to st for U. S. gal (liq). Putting the nubers into the teplate doing the arithetic then gives 9 9 gal = gal 1.609344 1 gal 1 3.785306 L ( 9) ( 1.609344) = = 1.395 3.785306 L L Rounded to one decial place then, the requested fuel consuption rating is 1.3 /L. Exaple 6: A particularly fine variety of cheese is sold for $1.47 per ounce. What is this price in dollars per kilogra? What we need to do here is to convert ounces to kilogras, since the other unit, dollars, is the sae in both the original value the requested value. Fro the table of unit conversion factors, we have 1 pound = 1 lb = 16 oz = 16 ounces 1 lb = 453.5937 g we know how to convert fro gras to kilogras. Thus, an effective strategy appears to be oz lb g kg The teplate for this calculation is $ $ oz lb g 1.47 = 1.47 oz oz lb g kg Putting the nubers into this teplate then doing the arithetic gives David W. Sabo (003) More Unit Conversion Exaples Page 5 of 1
$ $ 1.47 = 1.47 oz oz 16 oz 1 lb 1 lb 453.5937 g 1000 g 1kg ( ) ( ) ( ) 1.47 16 1000 $ $ = = 51.857 453.5937 kg kg Since aounts of oney are norally rounded to two decial places, we state our final answer as: the price of this cheese is $51.85 per kilogra. Exaple 7: Horse race lengths are oen easured in furlongs, where 8 furlongs is equivalent to one le. A fairly fast horse can run a 9 furlong race in about 1 nute 45 seconds. Copute the speed of this horse in /h. Before attacking the ain proble, we need to recognize that the tie interval stated here has xed units, nutes seconds, we ust convert it to one of these units alone, or the other alone. For now, the easiest is to convert to seconds: 60 s 1 nute 48 seconds = 1 n + 48 s = 108 s 1 n Thus, the horse s speed can initially be written as 9 furlongs 9 furlongs or 108 s 108 s Now we can set up the required unit conversion calculation. Since 1 = 8 furlongs 1 = 1.609344 1 h = 60 n 1 n = 60 s, then the two required stages of the unit conversion calculation can be accoplished with the strategies: furlong s n h The teplate for the calculation is 9 furlongs 9 furlongs s n = 108 s 108 s furlong n h Putting in the nubers gives David W. Sabo (003) More Unit Conversion Exaples Page 6 of 1
9 furlong 9 = 108 s 108 furlong s 1 8 furlong 1.609344 60 s 60 n 1 1n 1h ( 9) ( 1.609344) ( 60) ( 60) ( 108)( 8) = = 60.3504 h h Thus, this horse would be averaging a speed of 60.35 /h, rounded to two decial places, over the duration of this race. Exaple 8: A rectangular region of forest easuring 5.6 by 9. was destroyed by fire. Copute the area of this region express your final answer in units of hectares. Just applying the forula for the area of a rectangle gives A = LW = (5.6 )(9. ) = 51.5 as the area of the region. What we ust do now is convert this area to units of hectares. We have that 1 hectare = 10 000 1 = 580 1 = 30.48 c we know how to convert centietres to etres. Thus a prosing strategy to use here is c each repeated twice (because we have two units of to begin with, which ust eventually be converted to give in total) then hectare (ha) The required teplate of calculations is c ha = 51.5 51.5 With nubers inserted, this gives c 580 30.48 c 1 1ha = 51.5 51.5 1 1 100 c 10000 David W. Sabo (003) More Unit Conversion Exaples Page 7 of 1
( 51.5) ( 580) ( 30.48) ( 100) ( 10000) = ha = 13343.6187 ha Rounding to the nearest whole nuber, the area of the burned region is 13344 hectares. Exaple 9: The acceleration due to gravity near the earth s surface (denoted conventionally by the sybol g) is 9.81 /s. This eans that the speed of a falling object (in the absence of air resistance) will increase by 9.81 /s for each second of fall. Convert this value into units of /s, units of /h/s. The two calculations requested here are relatively independent. First, to carry out the conversion fro /s to /s, we note that the tie units stay the unchanged, so we need siply to convert the length units fro etres to feet. This is easily done using the strategy c The teplate for the calculation is c 9.81 = 9.81 s s c With the nubers in, the final answer obtained is 9.81 9.81 s s = 100 c 1 1 30.48 c ( 9.81)( 100) 3.185 ( 30.48) s s = = Since the value of g we were given has three significant figures, we should probably round our final answer here to three significant figures, stating it as 3. /s. The second conversion requested is a bit ore coplicated. The length unit ust be converted fro etres to kiloetres, which is easy by itself. Also, one of the tie units ust be converted fro seconds to hours. Thus, the following strategies are required: s n h The required teplate for the calculation is s n 9.81 = 9.81 s s n h Putting the nubers in, we get David W. Sabo (003) More Unit Conversion Exaples Page 8 of 1
9.81 9.81 s s = s 1 1000 60 s 60 n 1n 1h ( 9.81) ( 60) ( 60) = = 35.316 1000 sh i sh i Rounding to three significant figures again, we state the final answer as 35.3 /h/s. The units written this way indicate the eaning of this nuber a bit ore clearly. When an object falls without air resistance near the surface of the earth, its speed will increase by 35.3 /h for each second that it falls. Exaple 10: Typical baroetric pressure as reported in Canada is 101 kpa = 101,000 N/. Convert this value to its equivalent in pounds per square inch, the units used for baroetric pressure in the United States. In our table of unit conversion factors, we have 1 N = 0.480894 lb 1 in =.54 c we know how to convert fro centietres to etres. Thus, an effective strategy here is the pair of transforations N lb c in The second path in the line above ust be executed twice, because the length units are squared in the initial quantity. The teplate required for this conversion is N N lb c 101000 = 101000 N c in Now, insert the nubers do the arithetic: N N 101000 101000 = 0.480894 lb 1 1 N 100 c.54 c 1in ( 101000) ( 0.480894) (.54) ( 100) lb = = 14.64881131 in lb in Rounding off to three significant figures, we get the final answer of 14.6 lb/in. David W. Sabo (003) More Unit Conversion Exaples Page 9 of 1
Exaple 11: It is estiated that approxiately 1.05 3 of rock was blasted into the atosphere as dust during the ajor eruption of Mt. St. Helens in May, 1980. To how any truckloads of 1 3 each does this volue of rock correspond? Obviously if we knew the volue of rock in units of 3, this would be quite an easy proble to solve. Thus, we start by converting 1.05 3 to its equivalent in cubic etres. We know that 1 = 580 1 = 30.48 c Thus, an appropriate strategy for carrying out this unit conversion is c This will have to be repeated three ties, since three units of les ust be converted to three units of etres. The required teplate for the calculation is 3 3 3 3 3 c 1.05 = 1.05 c Putting the nubers in, we get 1.05 = 1.05 3 3 3 580 30.48 c 1 1 3 1 100 c 3 ( 1.05) ( 580) ( 30.48) 3 ( 100) 3 3 = = 4,376,590,917 3 3 This nuber is so big that we ve broken the SI rule against using coas in nubers in writing it out. To get the nuber of truckloads, we just have to divide this nuber by 1 (which gives 364,715,909.7) then round up to the nearest whole truckload. Thus, the rock blown into the atosphere as dust during the eruption was equivalent to 364,715,910 truckloads of 1 3 each. (Actually, since the original estiate of the volue of this rock was given to just three significant figures, we should probably round our final answer off to three significant figures as well, stating it as 365,000,000 truckloads that 365 llion truckloads!) Exaple 1: Certain types of beverages (not Kool-Aid ) are typically sold in bottles containing a volue called a fih, which is actually one-fih of a U. S. liquid gallon (or four-fihs of a U. S. liquid quart). Deterne how any llilitres of beverage this volues equals. This proble can be solved quite quickly. We have the unit conversion factor 1 U. S. gal (liq) = 3.785306 litres David W. Sabo (003) More Unit Conversion Exaples Page 10 of 1
in our table of unit conversion factors. Thus, there is a very siple strategy here: U. S. gal (liq) litre llilitre Abbreviating U. S. gal as siply gal here, the required teplate for this calculation is 1 1 L l gal = gal 5 5 gal L Putting in the nubers then gives 1 1 gal = 5 5 gal 3.785306 L 1 gal 1000 l 1 L 1 = ( 3.785306 )( 1000 ) l = 757.061 l 5 Thus, a fih of, uh, beverage, is exactly 757.061 l. (This probably explains why certain types of beverage coonly coe in containers of 750 l.) Exaple 13: Astronoers easure certain distances in units of au or astronocal units, which are defined to be 9,955,807 les the average distance between the centers of the earth the sun. They define a second unit called the parsec, which is equal to 06,64.4 au. (There is a reason for this value being chosen, but we don t want to get into too any technicalities here.) Now, a parsec is a pretty fair distance, but when we get to talking about intergalactic space, it s ore convenient to use units of kiloparsecs. The question is this: how any kiloparsecs is there in a zettaetre? If your first answer was who cares? you are not exhibiting the right attitude of adventure challenge this proble represents. This is quite a coplicated question, it deals with units of easureent that few of us use in everyday conversation. Nevertheless, the unit conversion ethod we ve been illustrating throughout this docuent is up to the task of solving even this proble. Start by suarizing the inforation we have so far: 1 au = 9,955,807 1 parsec = 0664.4 au 1 kiloparsec = 1000 parsec 1 zettaetre = 1 Z = 10 1 etre Now, we need a strategy to convert fro Z to kiloparsecs. A bit of thought suggests the following: Z au parsec kiloparsec The teplate required here is now rather easy to write out, even if it is also rather lengthy: David W. Sabo (003) More Unit Conversion Exaples Page 11 of 1
au parsec kparsec 1Z = 1Z Z au parsec Putting the nubers in gives: 1 10 1 1 1au 1Z = 1Z 1Z 1000 1.609344 9,955,807 1parsec 1kparsec 0664.4 au 1000 parsec ( 10 1) = kiloparsecs ( 1000)( 1.609344)( 9,955,807)( 0664.4)( 1000) = 3.40786 kiloparsecs Thus, 1 Z is equivalent to 3.41 kiloparsecs, rounded to two decial places. Even though you probably have no ental iage of how big several of these units of length really are, you should be able to set up the calculations to coplete the conversion be able to confir that no errors have been ade in the calculation. David W. Sabo (003) More Unit Conversion Exaples Page 1 of 1