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Introduction to Dimensional Analysis

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Introduction to Dimensional Analysis Introduction to Dimensional Analysis Dimensional analysis is a math tool used to change units. With this method, it is easy to change very complex units provided you know the necessary conversion definitions. This method is also referred to as unit conversion, factor analysis, unit analysis, DA, factor-label method, unit multipliers or unit fractions. Here are some basic concepts that are used in dimensional analysis: Any whole number can be written as a fraction. A number can be changed to a fraction and still keep its value by placing 1 as the denominator. 4 Ex. 4 1 The value of a quantity is not changed when it is multiplied by 1. When a number is multiplied by 1, the value of the number remains unchanged. Ex. 12(1) = 12 One (1) can be written as a fraction in many ways. As long as the numerator of the fraction equals the denominator of the fraction, the fraction is equal to one. 1 2 7 45 1 1 2 7 45 Unit fractions are also fractions that equal 1. The difference is these fractions are in units. Any definition involving units can be made into a unit fraction. 12 inches = 1 foot 12 inches 1 foot 1 1 foot 12 inches The unit labels are crucial to forming unit fractions. If the labels are removed, the statement is no longer true. Since unit fractions equal 1, it is possible to multiply a quantity by a unit fraction and not change the value. Unit labels cancel in the same way that common factors reduce to 1 in fraction multiplication. 4 5 4 1 yard 1 foot 1 yard 5 7 7 3 feet 12 inches 36 inches Unit fractions are used to convert the units of a given number while retaining the number’s value. Remember, the unit fraction equals one. When a number and a unit fraction are multiplied, the number retains its value because any number that is multiplied by one stays the same. The only change will be in the units assigned to the value. When a measurement is changed using unit fractions, the appearance of the measurement changes but not its size. Units of Capacity Units of Length 1 gal = 4 qt 1 kL = 1 daL 1 ft = 12 in 1 in = 2.54 cm 1 cm = 10 mm = 8 pt 1 dm = 10 cm 1 qt = 2 pt 1 qt = .9463 L 1 hL = 10 daL 1 yd = 3 ft 1 yd = .9144 m 1 m = 10 dm 1 pt = 2 cu 1 daL = 10 L = 36 in = 100 cm 1 cup = 8 oz 1 fl oz = 1 L = 1000 mL = 1000 mm = 16 Tbs 29.57 mL = 10 dL 1 mile =5280 ft 1 km = .621 mi 1 km = 1000 m 1 fl oz = 2 Tbs 1 dL = 10 cL Example #1: Change 4 miles into yards. 1. Write 4 miles as a fraction. 4 miles 1 2. Since there is not a definition that relates miles to yards, unit fractions are used to change miles into feet and then into yards. The unit fractions for miles and feet are 1 mile 5280 feet or . Because we do not want miles in the final answer, we 5280 feet 1 mile multiply by the unit fraction that will allow us to cancel the mile units. 4 miles 5280 feet 4 miles = 1 1 mile We do not want feet in the final answer so we find the unit fractions that will allow us to 1 yard 3 feet cancel feet: or . We use the first one. 3 feet 1 yard 4 miles 5280 feet 1 yard 4 miles = 1 1 mile 3 feet 3. Now that we have the desired units (yard), we multiply all of the numerators together and all of the denominators together. 4 miles 5280 feet 1 yard 21120 yards 4 miles = 1 1 mile 3 feet 3 To get the final result, divide the numerator by the denominator. 4 miles 5280 feet 1 yard 21120 yards 4 miles = 7040 yards 1 1 mile 3 feet 3 Example #2: Change 640 fluid ounces into pints. 640 ounces 1 cup 1 pint 640 pints 640 ounces 40 pints 1 8 ounces 2 cups 16 The above examples involve changing one unit to another. Some problems are more complicated than that. For these we use as many unit fractions as we need, setting up a long multiplication, so the units we don’t want cancel out. Units of Time Units of Weight Units of Area 1 millennium = 1000 yrs 1 tonne = 1 ft2 = 144 in2 1 in2 = 6.452 cm2 1 T = 2000 lb 1 cent = 100 yr =2205 lb 1 yd2 = 9 ft2 1 yd2 = .8361 m2 1 yr = 12 mo 1 lb = 16 oz 1 lb = .4536 kg 1 acre = 160 rd2 1 a =.4047 ha = 365 days 1 oz = 16 dram 1 oz = 28.35 g = 4840 yd2 1 day = 24 hrs 1 dram = 1 dr = 1.772 g 1 mi2 = 640 1 hr = 60 min 27.34 grain acres 1 min = 60 sec Example #3: Which is faster: 80 miles an hour or 40 meters per second? To compare two values, they must have the same units. Using the definitions given in the preceding tables, it is possible to change miles to feet to inches to centimeters to meters and hours to minutes to seconds. If the tables gave different definitions, we could choose different unit fractions. We use whatever definitions we have available. 1. We begin by changing one set of units. Let’s start with the miles. 80 miles 5280 feet 12 inches 2.54 cm 1 meter hour 1 miles 1 foot 1 inch 100 cm The units for this problem are now meters per hour. 2. To finish the unit change, we multiply by the unit fractions that will change hours to seconds. 80 miles 5280 feet 12 inches 2.54 cm 1 meter 1 hour 1 minute hour 1 miles 1 foot 1 inch 100 cm 60 minutes 60 seconds The time units have now changed from hours to seconds and the units for the problem are meters/second. 3. Now that we have the desired units, we multiply all of the numerators together and all of the denominators together. 80 miles 12,874,752 meters hour 360,000 seconds To get the final result, divide the numerator by the denominator. 80 miles 12,874,752 meters 35.76 meters hour 360,000 seconds second This tells us that 80 miles per hour is equivalent to a little less than 36 meters per second, so 40 meters per second is faster than 80 miles per hour. Example #4: You are planning a trip to Mexico and, in the process, you find out the cost of gasoline there is 7 peso per liter. How does that compare to the cost of gas in the US? ($1 = 12 peso) 7 peso 1 dollar .9463 liter 4 quarts 26.50 dollars 2.21 dollars 1 liter 12 peso 1 quart 1 gallon 12 gallons gallon 7 peso per liter is about the same as $2.21 per gallon. Example #5: You are installing a fence and you need to mix some concrete to set the posts. The directions say you need five gallons of water for your mix. Your bucket isn’t marked so you don’t know how much it holds. However, you just finished a two-liter bottle of soda. If you use the bottle to measure your water, how many times will you need to fill it? 5 gallons 4 quarts .9463 liters 1 bottle 18.93 bottles 9.46 bottles 1 bag of mix 1 gallon 1 quart 2 liters 2 bag of mix 1 bag of mix You need to use about 9 1/2 bottles of water. Suggestions for using Dimensional Analysis 1. Determine what you want to know. (How many miles in an hour?) Rephrase it using “per”. (miles per hour) 2. Determine what you already know. 3. Determine the unit fractions that may be needed. You will need enough to form a “bridge” from what you know to what you want to know. 4. From what you know, pick a starting factor. 5. Select a unit fraction that cancels out units in the starting factor that you don’t want. 6. Continue picking unit fractions until only the desired units remain. 7. Do the math. Multiply the numerators together. Multiply the denominators together. Divide the numerator by the denominator. Be sure to write the labels that go with the answer. 8. Take a few seconds and ask yourself if the answer makes sense. If it doesn’t, check your work. .
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