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Understanding Dimensional Analysis: Using Conversion Factors for Measurement Conversion

Dimensional analysis, also known as factor-labeling, is a powerful technique that utilizes conversion factors to convert measurements from one unit to another while maintaining numerical values. This method requires expressing all measurements with both numerical values and their corresponding units. For example, understanding that 1 gallon equals 4 quarts allows us to convert units effectively. Further, dimensional analysis can simplify problems such as converting miles to kilometers, where knowing that 1 km equals 0.621 miles helps in determining distance equivalences efficiently.

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Understanding Dimensional Analysis: Using Conversion Factors for Measurement Conversion

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  1. Dimensional Analysis (aka Factor-Label) This technique involves the use of conversion factors and writing all measurements with both numerical values and the unit of measurement A conversion factor is where you have the same amount (entity) represented by two different units of measurement with their corresponding numerical values

  2. Conversion Factors • Here are some examples • 1 foot = __ inches • 1 kilometer = ____ meters • 1 inch = 2.54 centimeters • 1 gallon = __ quarts • 1 acre = 4840 square yards • 1 day = ___ hours 12 1000 4 24

  3. Conversion factors…cont. Conversion factors for 1 ft = 12 in There are almost an infinite number of conversion factors that include meters:

  4. Conversion Factors….cont. • One member of a dinner party orders a 16 ounce steak and another orders a one pound steak- Compare the two steaks • They are the same since 16 oz dry wt. = 1 pound

  5. Conversion Factors….cont. • In grade school we learned that 1 gallon contained 4 quarts or stating that relationship as an equality: • 1 gallon = 4 quarts • Since 1 gallon and 4 quarts represent the same amount, we have a Conversion Factor

  6. Conversion Factors….cont. • Start with 1 gallon = 4 quarts • Dividing each side by 1 gallon we get this equation • 1 gallon = 4 quarts 1 gallon 1 gallon Since 1 gallon divided by 1 gallon equals 1 • Our equality becomes:1 = 4 quarts 1 gallon

  7. Conversion Factors….cont. • Again start with 1 gallon = 4 quarts • But this time we’ll divide each side of the equality by 4 quarts • The resulting equation is • 1 gallon = 4 quarts 4 quarts 4quarts

  8. Conversion Factors…. Cont. • The right side of our equation becomes one because 4 quarts divided by 4 quarts is 1 • 1 gallon = 14 quarts • Rearranging this becomes 1 = 1 gallon 4 quarts

  9. Conversion Factors….cont. • A mid-presentation summary • We know that 1 gallon = 4 quarts • Using a little mathematical magic • 1 gallon = 1 and 4 quarts = 1 4 quarts 1 gallon • Why is this an important concept?

  10. Conversion Factors….cont. • Now a little math review……………. • What is 5 x 1? • What is 5 x 2 ? 2 • Both expressions give you the same answer- why? • Because 2/2 equals 1 and therefore the second equation is just like the first and we did not change the initial value of 5.

  11. Putting It TogetherHere’s An Example • How many quarts are in 15 gallons ? • Remember we do NOT want to change the amount represented by 15 gallons, only the units in quarts • So we’ll use the conversion factor between gallons and quarts; that is 1 gallon = 4 quarts

  12. Our Example continued……. • We set it up like this: 15 gallons x 4 quarts 1 gallon • Cancel units • Do the math to complete the problem • 15 x 4 quarts = 60 quarts

  13. Every measurement must have a unit. 60 quarts

  14. What do I need to do? • From the problem determine the following: • Known quantity (number and units) which is called the Given • Identify what the Desired units are • Conversion factor(s) needed (both universal and question specific)

  15. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km First write down the desired quantity

  16. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Next, equate desired quantity to the given quantity

  17. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi Now we have to choose a conversion factor

  18. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Pick the one that will allow you to cancel out miles

  19. 1 km 0.621 mi 0.621 mi 1 km Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Multiply given quantity by chosen conversion factor

  20. x 1 km 0.621mi Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47mi Cross out common factors

  21. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Cross out common factors

  22. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) #km = 47 Are the units now correct?

  23. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) #km = 47 Yes. Both sides have km as units.

  24. x 1km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 Yes. Both sides have km as units. #km

  25. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) = 75.7 km # km = 47 Now finish the math.

  26. x 1 km 0.621 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) = 75.7 km # km = 47 The final answer is 76 km (correct sig fig)

  27. Summary The previous problem was not that hard. In other words, you probably could have done it faster using a different method. However, for harder problems the factor label method is easiest.

  28. Let’s answer the beginning questions • The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s. • Which one is faster? • Both must be in the same units, so we must convert one. • Does it matter which one? • NO.

  29. Factor label example Question: 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m/s First write down the desired quantity

  30. Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m/s = 27 mi/hr Next, equate desired quantity to the given quantity

  31. Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m = 27 mi x s 1 hr Now we have to choose conversion factors

  32. 1 mi 1.609 km 1 mi 1.609 km Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m = 27 mi s 1 hr Pick the one that will allow you to cancel out miles

  33. 1.609 km 1 mi Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m = 27 mi X s 1 hr Multiply given quantity by chosen conversion factor

  34. 1.609 km 1 mi Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m = 27 mi X s 1 hr Cross out common factors

  35. 1.609 km 1 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec #m = 27 X s 1 hr NO, both sides aren’t equal Are the units now correct?

  36. 1.609 km 1000 m 1 km 1 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec # m = 27 X X s 1 hr Must choose another factor Cross out common factors

  37. 1.609 1000 m 1 1 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec #m = 27 X X s 1 hr Do units match? NO, must choose another factor

  38. 1.609 1000 m 1 hr 60 min 1 1 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec #m = 27 X X X s 1 hr Cross out common factors Do units match

  39. 1.609 1000 m 1 60 min 1 1 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec #m = 27 X X X 1 s Do units match? Cross out common factors NO, must choose another factor NO – must choose another factor

  40. 1000 m 1 min 1.609 1 1 60 s 1 60 min Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec X #m = 27 X X X s 1 NO, must choose another factor Cross out common factors NO – must choose another factor Do units match? Cross out common factors

  41. 1000 m 1 1.609 1 1 60 s 1 60 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec X #m = 27 X X X s 1 NO, must choose another factor Cross out common factors NO – must choose another factor Do units match? Do units match?

  42. 1 1.609 1 1000 m 1 60 s 1 60 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec X #m = 27 X X X s 1 Do units match? NO, must choose another factor Cross out common factors NO – must choose another factor Do units match? YES !

  43. 1.609 1 1 1000 m 1 60 s 1 60 Factor label example Q – 27 mph is equal to how many m/s? factors needed: 1 mi = 1.609 km 1 hr = 60 min 1000 m = 1 km 1 min = 60 sec X #m = 27 X X X s 1 = 12.0675 m/s Do the math = 12 m/s (correct sig fig)

  44. The fastest human is reported to be able to run at a rate of 27 mph, while the fastest fish can swim at a rate of 31 m/s. Which one is faster? How much faster? • Human: 27 mph = 12 m/s • Fish: 31 m/s • Which one is fastest? • How much faster? 31m/s – 12 m/s = 19 m/s

  45. Working with metric/SI quantitybase unit • length meter • mass gram • volume liter

  46. SI Base Units

  47. Working with metric/SI Base Unit gram meter liter m

  48. Working with metric/SI When converting within the metric system it is helpful to remember: • “1 always goes with the prefix” • the value of the prefix goes with base unit

  49. Conversions with metric/SI • Example: How many meters are in 12 km? # m = 12 km x 1000 m m m = 1.2 x 104m 1 km Base Unit gram meter liter Use chart to get conversion

  50. Conversions with metric/SI • Example: How many cm are in 1.3 m? # cm = 1.3 m x 1 cm 0.01 m Base Unit gram meter liter Use chart to get conversion

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