Dimensional Analysis #3


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1 Dimensional Analysis #3 Many units of measure consist of not just one unit but are actually mixed units. The most common examples of mixed units are ratios of two different units. Many are probably familiar to you. For example, you drive your car at a speed of 55 miles per hour. You get gas mileage of 28 miles per gallon. You pay $.38 per gallon for gasoline. You inflate your car tires to a pressure of 35 pounds per square inch. You drive an average of 2,000 miles per year. Each of these mixed units involves two different types of units. We can write each of these mixed units as a unit fraction. For example, a speed of 55 miles per hour could be written as or Either of these two ratios expresses the relationship between miles and hours; that is, that in one hour the car will travel a distance of 55 miles. Some other mixed units are products of two or more units. For example, torque is measured in footpounds, electricity usage is measured in kilowatthours, and production is measured in manhours. In each of these mixed units two different units are multiplied together, but the unit is generally written with a hyphen; that is, you will usually see footpounds not foot pounds or foot pounds. We can use the method of dimensional analysis to convert from one mixed unit to another mixed unit of similar type. To perform a conversion on a mixed unit expressed as a ratio, we will often have to chain several unit fractions to eliminate units in both the numerator and the denominator of the fraction. EXAMPLE : The speed limit on many highways is 55 mph. Convert this speed to feet per second. SOLUTION: First, note that the mixed unit we are given, 55 mph or 55 mi/hr, has a unit of distance in the numerator and a unit of time in the denominator. We can convert this mixed unit to any other mixed unit which has a unit of distance in the numerator and a unit of time in the denominator. The unit of ft/sec is of this type. To perform the conversion, we must multiply by unit fractions as we did in the previous two sections. We want to choose unit fractions so that the unit of miles will cancel and we are left with feet in the numerator. We will also want to multiply by other unit fractions so that the unit of hours will cancel and we will be left with the unit of seconds in the denominator. We can eliminate either the miles or the hours first. In this example, we will first eliminate the miles by multiplying by a unit fraction that has miles in the denominator ft mi If we were to multiply these two fractions, the result would be a fraction with feet in the numerator and hours in the denominator (ft/hr). Next we must multiply by a unit fraction or fractions that will eliminate hours and yield seconds. Since there are 60 seconds in minute and 60 minutes in hour, we will multiply by two unit fractions to convert from hours to seconds ft mi 60min min 60sec
2 Dimensional Analysis #3, Continued When we multiply these four unit fractions together, miles, hours and minutes will cancel, and we will be left with ft/sec, which is the unit we wanted. 55 mi 5280 ft mi 60min min 55(5280)()() ft = ft sec 60sec ()(60)(60)sec Therefore, a speed limit of 55 mph is equivalent to about ft/sec. REMEMBER: To convert from one mixed unit in fraction form to a different mixed unit in fraction form, you may eliminate the units in the numerator or in the denominator first. Just multiply by as many unit fractions as are necessary to convert both. EXAMPLE 2: A chemistry book gives the density of copper as 8.94 g/cm 3. What would this density be if expressed in kg/m 3? SOLUTION: The density of copper can be written as the ratio 8.94g cm 3 We must convert from grams to kilograms and from cubic centimeters to cubic meters. It makes no difference which conversion we perform first. For this example, we will first convert from grams to kilograms by multiplying by a unit fraction with grams in the denominator. 8.94g cm kg 3 000g Next we will convert from cubic centimeters to cubic meters by multiplying by a unit fraction with cubic centimeters in the numerator. Remember that since these are cubic units, we must use the 00 cm conversion factor of three times. m 8.94g cm kg 3 000g (00)3 cm 3 m 3 Now when we multiply these fractions together, the units of grams and cubic centimeters will cancel, and we will be left with the desired units of kilograms per cubic meter g cm 3 kg 000 g (00)3 cm 3 m 3 = 8940 kg m 3 Therefore, the density of copper is 8940 kg/m 3. The technique of dimensional analysis can be used to solve many problems that might be solved by formulas or by other methods. If we know what unit is required for an answer, we can multiply by the right unit fractions to get that unit. The next problem gives an example of this method. 2
3 Dimensional Analysis #3, Continued EXAMPLE 3: Suppose that a bathroom faucet is dripping at a rate of 40 drops per minute. At this rate, how long would it take to drain a 40gallon water tank. (We will use the fact that 5 drops ml) SOLUTION: Since the volume of water in the tank is expressed in gallons and the dripping of the faucet is expressed in terms of drops per minute, we must convert the gallons to drops or the drops per minute to gallons per minute. For this example, we will convert drops/min to gal/min as shown below. 40drops min ml 5drops L 000 ml.057qt L gal 4qt gal min This new rate can be expressed as a unit fraction in two different ways gal min or min gal We can use the second fraction to determine the time it would take to drain the water tank in minutes. 40gal min gal We can multiply by still more unit fractions to express the time in hours, days, or any other unit of time. We will calculate the time in hours, rounding to the nearest whole number. 40gal min gal 60min 946hr Therefore, at the given rate, it would take approximately 946 hours of continuous dripping for the 40gallon water tank to be drained. 3
4 PROBLEM SET 3 Answers to the oddnumbered problems are given at the end of the problem set. Dimensional Analysis #3, Continued WARMUP EXERCISES Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places.. 30 m/sec to km/hr 2. 7 kg/m 2 to lbs/ft ft/min to cm/sec 4. 3 cents/ft to cents/cm g/cm 3 to lb/in mi/gal to km/l 7. $90/day to cents/min 8. 2 ft 3 /sec to gal/min PROBLEMS Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. 9. Gasoline costs $.469 per gallon. How many cents per liter does the gas cost? Round your answer to the nearest cent. 0. The national debt is about $3,233,000,000,000. If a person can count one dollar per second, how many years would it take to count the number of dollars in the national debt? (Assume that the person doing the counting takes no break in counting.) Round your answer to the nearest year.. A farmer has irrigation rights of 2 acrefeet per year. One acrefoot is the amount of water that would cover one acre to a depth of one foot. How many gallons of water would the farmer have to pump to use all of the allotted irrigation water in one year? Round your answer to the nearest thousand gallons. 2. A bicycle tire is inflated to 20 psi (lbs per square inch). What would this pressure be in Newtons per square meter? Round your answer to the nearest hundred. EXTRA PROBLEMS Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places m/sec to in/sec gal/min to ft 3 /sec cm 3 /min to mm 3 /sec /in. to $/yard 4
5 Dimensional Analysis #3, Continued Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. 7. A bolt must be tightened to 60 ft lbs of torque. To the nearest tenth, what is this torque in N m? SOLUTIONS TO ODDNUMBERED PROBLEMS: (NOTE: For some of these problems there are several ways to set up the problem. Therefore, your unit fractions may look different from the sequence of unit fractions shown here. Your final answer, however, should be approximately the same as the one given below.) The gasoline costs about 39 cents/liter.. The farmer must pump 489,000 gallons of water to use all of the allotted irrigation water The bolt must be tightened to about 8.3 N m of torque. ANSWERS TO EVEN NUMBERED PROBLEMS lbs / ft / cm km / L gal / min 0. It would take 02,57.76 years psi is approximately 827,000 N / m ft 3 /sec $ / y 5
6 MEASUREMENT AND CONVERSION TABLE U. S. CUSTOMARY SYSTEM yd = 3 ft 3 tsp = T ft = 2 in 6 T cup fathom = 6 ft cup = 8 oz (liquid capacity) mi = 5, 280 ft pt = 2 cups acre = 43,560 ft qt = 2 pt lb = 6 oz (dry weight) gal = 4 qt ton = 2000 lb gal 23 in ft 7.48 gal METRIC SYSTEM m =,000,000 microns ( ) hectare (ha) = 0,000 m m = 000 mm kg = 000 g m = 00 cm g = 000 mg m = 0 dm kl = 000 L km = 000 m L = 000 ml cm = 0 mm cm = ml m = 000 L CONVERSION BETWEEN THE U. S. CUSTOMARY AND THE METRIC SYSTEM in. = 2.54 cm lb g m in. oz g mi.609 km kg lb kwh = 3,43 Btu pt cm lb. = N (Newtons) L.057 qt tsp = 5 ml ft L
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