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Dimensional Analysis (aka Factor-Label)

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

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Dimensional Analysis (aka Factor-Label)

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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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