Handout Unit Conversions (Dimensional Analysis)

1 1 Handout Unit Conversions (Dimensional Analysis) The Metric System had its beginnings back in 1670 by a mathematician called Gabriel Mouton. The modern version, (since 1960) is correctly called "International System of Units" or "SI" (from the French "Syst me International"). The metric system has been officially sanctioned for use in the United States since 1866, but it remains the only industrialized country that has not adopted the metric system as its official system of measurement. Many sources also cite Liberia and Burma as the only other countries not to have done so. This section will cover Conversions (1) selected units in the metric and American systems, (2) compound or derived measures, and (3) between metric and American systems. Also, (4) applications using Conversions will be presented. The metric system is based on a set of basic units and prefixes representing powers of 10. Prefixes (The units that are in bold are the ones that are most commonly used.) prefix symbol value giga G 1,000,000,000 = 109 ( a billion) mega M 1,000,000 = 106 ( a million) kilo k 1,000 = 103 hecto h 100 = 102 deca da 10 1 deci d 101= = 10-1 centi c 1001= = 10-2 milli m 000,11= = 10-3 (a thousandth) micro (the Greek letter mu) 11, 000, 000= = 10-6 (a millionth) nano n 000,000,000,11= = 10-9 (a billionth) Basic metric units length meter (m) mass gram (g) volume liter (L) time second (s) temperature Celsius ( C) or Kelvin ( K) where C= 2 To get a sense of the size of the basic units of meter, gram and liter consider the following examples.

2 The standard height of a door handle or knob is 1 meter. The weight of a penny is grams. Almost everyone is familiar with 2 liter bottles of soda. The next set of examples illustrates the use of some of the most used prefixes. Examples: 1. 1 milligram =11, 000gram = gram, or 1 mg =11, 000 g = gram 2. 1000 mg=1g 3. 1 milliliter=11, 000liter= liter, or 1 mL=11, 000L= L 4. 1,000 mL = 1 L 5. 1 centimeter= 1100 meter= meter, or 1 cm =1100 m = m 6. 1 kilogram = 1,000 grams, or 1 kg = 1,000 g 7. 1 kilometer = 1,000 meters, or 1 km = 1000 m 8. 1 kilobyte = 1000 bytes (a byte is a unit used in computers to express the length of a word .) 9. 1 megabyte = 1,000,000 bytes = 1000 kilobytes 10. An adult male is about how tall? A) 175cm B) 175m C) 175mm The correct answer is A) since he is almost twice the standard height of a door knob, so almost 2 meters, that is almost 200 cm. 11. Choose the most reasonable weight of a newborn baby: A) B)13L C) 4kg D) The correct answer is C) since is like a penny, while L measures volume and m is for length.

3 3 conversion Tables The American system of measurements is based on the English system. American-->AmericanCapacity (volume)1 gallon = 1gal = 4 quarts = 4 qts1 quart = 1 qt = 2 pints = 2 pt1 pint = 1 pt = 2 cups = 2 c = 16 fluid ounces = 16 fl oz1 cup = 1c = 8 fluid ounces = 8 fl. oz (Note: a fluid ounce is not the same as the weight measure ounce.)1 fluid ounce = 1 fl oz = 2 tablespoons = 2 tbsp1 table s poon = 1 tbsp = 3 teaspoons = 3 tspLength 1 foot = 1 ft = 12 inches = 12 in1 mile = 1 mi = 5280 feet = 1760 yd Mass1 pound = 1 lb = 16 ounces = 16 oz1 ton = 2000 pounds = 2000 lb (This is sometimes called a short ton or ton(US). A long ton or ton(UK) is equal to 2240 pounds.)TemperatureTemperature is measured in degrees Fahrenheit ( F). Water freezes at 32 F and boils at 212 FSI--> American1 meter = yards = feet = inches1 kilometer = .62137 miles1 gram = .03527 ounces1 kilogram = ounces = pounds1 liter = .26417 gallons = quartsSI temperature measure is degree Celsius ( C).

4 Water freezes at 0 C and boils at 100 F Another SI measure for temperature is Kelvin (K) where American--> SI1 inch = meters = centimeters1 yard = meters1 mile = meters = kilometers1 pound = grams1 ounce grams1 gallon liters1 quart .9464 liters1 fluid ounce .02957 liters = milliliters1 tsp = mL 59 32CF Everywhere in the WorldTime1 minute = 1 min = 60 seconds = 60 s1 hour = 1 hr = 60 min1 day = 24 hrSmall amount of liquid Milliliters are used extensively in the sciences and medicine. The abbreviation cc is often used instead of mL. The relation of mL to volume is given by4 Areas and volumes Recall that area is measured in squares. One square foot or 1 ft2 represents the area covered by a square 1 foot by 1 foot. Example: 12 square feet = 12 ft2 means 12 squares 1 foot by 1 foot. The measure of the area of a rectangle 3 ft by 4 ft is 12 ft2 since it can be covered (tiled) with 12 squares each 1 foot by 1 foot. Volume is measured in cubes.

5 One cubic centimeter or 1 cm3 represents the volume that can be covered by a cube 1 cm by 1 cm by 1 cm. Example: Find the volume of a box 3 cm by 3 cm by 4 cm. The question is, how many cubes 1 cm on a side can be put into this box. 3 cm 3 cm 4 cm = 36 cm3. That is, the box will hold 36 cubes 1cm by 1 cm by 1 cm. A conversion table could be made for areas and another could be made for volumes. The tables would include facts like 1 ft2 = 144 in2, 1 yd2 = 9 ft2, 1 m2 = 10000 cm2, 1 ft3 = 1728 in3, etc. Fortunately, as shown below, this is not necessary. This Handout will cover one method commonly used in the sciences to convert units. This method uses multiples of 1" in a convenient form. Examples: a) 4000 in = ? ft Notice that 1 ft = 12 in. We write our 1" as112ftin so that in in 4000 in will cancel. 4, 00014, 0004, . notice that the in canceled b) 18 ft = ? in 18121821611ftinftinft Notice that 1 ft = 12 in. This time we write our 1 as 12in / 1ft so that the ft cancel leaving only in as the unit.

6 C) mi = ? in From the conversion tables on page 3 notice that 1 mi = 5280 ft and 1 ft = 12 in. So we should write our 1 in two different ways 1=5280 1 and as 1=12 1 . Using the first fraction we can convert miles into feet, while using the second fraction we can further convert the feet into inches. Combining these two we have the following: 1 2 3 4 5 6 7 8 9 10 11 12 5 12( )(5280)(12) notice that the mi and the ft cancel d) 146000 cm = ? km Just like in the previous example we don t have a direct conversion between centimeters (cm) and kilometers (km). However, both are connected through meters (m). Therefore, we will use the 1 in two different ways: 1=1 100 and 1=1 1000 . Using both, we get the following: (100)(1000)cmmkmcmkmkmcmm notice that the cm and the m cancel e) 1,000,000 fl oz = ? gal From the conversion tables on page 3, we see that 1 = 16 , 1 =2 and 1 =4 . We have to combine all three of these Conversions to go from fluid ounces to gallons: (16)(2)(4)fl ozptqt galfl ozgalgalfl oz ptqt notice that the fl oz , the pt , and the qt cancel f) 18 ft2 = ?

7 Yd2 The following is the standard method for handling areas and volumes. The conversion tables on page 3 say that 1 =3 . For square units, we have to use this conversion twice so that ft2" cancels with ft ft . 2222181118182133(3)(3)ftyd ydftydydftft Another way to obtain the same result is to create a new conversion formula for our specific prupose. Since 1 =3 we immediately have 1 2=3 3 =9 2 and 22222218181182119ftftydftydft g) 1,000,000 mm2 = ? km2 This time we will use two Conversions 1 m = 1000 mm and 1 km = 1000 m. Each of these two conversion will be used twice, since we are using square units: 22210000001110000001110001000mmmmmmmmmmm Then we continue: . Therefore 1000000 2= 2. notice that mm2" cancels with mm mm in the first row, and m2 cancels with m m in the second row of calculations above. h) 18 ft3 = ? yd3 When dealing with cubic units, we have to apply each conversion formula three times. For example, since 1 =3 , we can apply this conversion three times so that ft3 cancels with ft ft ft.

8 333318111182181333(3)(3)(3)3ftyd yd ydftydydftftft 6 Or, as in the example f) above, we can create our own conversion formula 33133327ydftftftft and use it directly: 333333318181182181127273ftftydftydydft i) 8 mL = ? cc This is a straight forward conversion , since 1 =1 : 818811mL ccmLccmL j) 3 L = ? cc This time we have to use two Conversions in a row: 1 L = 1000 mL and 1 mL = 1 cc: 3 10001(3)(1000)330001111 LmL ccLccccLml k) 1 m3 = ? cc For cubic units, we use conversion formulas three times: 33311001001001(100)(100)(100)11000000111 111mcmcmcm ccmccccmmmcm Alternatively, we can derive our own conversion formula for cubic units. Using 1 =100 , we get. 3311001001001000000mcmcmcmcm l) 74 F = ? C C = 95(F 32) C = 95(74 32) C C m) 29 C = ? F F = 59C + 32 F = [59(29) + 32] F = F Compound Measures We now look at what is sometimes called compound measures: miles per hour = mi/hr, ft/sec, lb/ft2, g/cm3 etc. Examples a) If a car is traveling 90 mi/hr, how many feet will the car travel in 1 sec?

9 In 5 sec? A direct way to convert these units is: 90528011min(90)(5280)90// sec 132 / sec1160 min 60 sec(60)(60)mifthrmi hrftfthrmi 7 Alternatively, we can derive our own conversion formulas separately for miles and for hours: 1 =5280 and 1 =60 =60 60 =3600 . We then replace miles with 5280feet and hours with 3600sec. Therefore 9090 5280132 /3600miftft sechrsec In 1 second the car will travel 132 ft. The car will travel (5 sec )(132 ft/sec) = 660 ft. in 5 seconds. b) A 90 lb weight is applied to an area 1/4 in by 1/4 inch. This pressure is equivalent to how many tons per ft2 ? We have 90 pounds per (1/4)(1/4) in2. We have [90 lb/(1/16) in2] = 1440 lb/in2. 22221440112 12(1440)(12)(12)1440 inton ftton ftinlbftft c) If a machine produces 225 parts in an 8 hour day, how many minutes will it take to produce 10 parts? We will first get minutes per part. 860 min8(60)32min/min/225122515hourspartpart partshour The time needed for 10 parts would be (10)(32/15) minutes = 64/3 minutes = 21 minutes.

10 D) If a machine produces 225 parts in an 8 hour day, how many parts can it produce in 10 minutes ? We will first get parts per minute. Notice that we have parts in the numerator since we want parts in the final answer. 225122515/ min/ min860 min8(60)32partshourpartspartshours In 10 minutes, the machine would produce 151032 parts = 7516parts =.11416 e) Water is pouring into a plastic box 1 m by 1 m by .3 m at the rate of 125 mL per minute. How long will it take in hours to fill this box? The volume of the box is 1 m 1 m .3 m = .3 m3. In 1 minute, 125 mL will drip into the box. We will convert mL to cm3 to m3 and minutes to hours. 3331min11001001001(100)(100)(100) min(125)(60)mLcmcmcmhrhrhrmLcmmmmmm hrs for each m3. It will take ( )(.3) hr = 40 hr to fill the box Homework: 1. 1000 yd = ? in 2. 2000 in = ? ft 3. 1000 in = ? yd 4. 2000 ft = ? in 5. 1000 yd2 = ? in2 6. 2000 in2 = ? ft2 7 1000 in2 = ? yd 8. 2000 ft2 = ? in2 9. 1000 yd3 = ? in3 10.