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

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

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

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

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

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

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

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

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?

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.

Putting It Together Here’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

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

Every measurement must have a unit. 60 quarts

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)

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

Next, equate desired quantity to the given quantity 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

Now we have to choose a conversion factor 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

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

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

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

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

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

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

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

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

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

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.

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.

First write down the desired quantity 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

Next, equate desired quantity to the given quantity 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

Now we have to choose conversion factors 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

Pick the one that will allow you to cancel out miles 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 1.609 km 1 mi 1 mi 1.609 km # m = 27 mi 1 hr s Pick the one that will allow you to cancel out miles

Multiply given quantity by chosen conversion factor 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 1 hr s 1.609 km 1 mi X Multiply given quantity by chosen conversion factor

Cross out common factors 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 1 hr s 1.609 km 1 mi X Cross out common factors

NO, both sides aren’t equal 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 1 hr s 1.609 km 1 X NO, both sides aren’t equal Are the units now correct?

Must choose another factor 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 1 hr s 1.609 km 1 1000 m 1 km X X Cross out common factors Must choose another factor

NO, must choose another factor 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 1 hr s 1.609 1 1000 m 1 X X NO, must choose another factor Do units match?

Cross out common factors 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 1 hr s 1.609 1 1000 m 1 1 hr 60 min X X X Do units match Cross out common factors

NO – must choose another factor NO, must choose another factor 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 1 s 1.609 1 1000 m 1 1 60 min X X X NO – must choose another factor Cross out common factors NO, must choose another factor Do units match?

NO – must choose another factor NO, must choose another factor 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 1.609 1 # m = 27 s X 1000 m 60 min 1 min 60 s X Cross out common factors NO – must choose another factor Do units match? Cross out common factors NO, must choose another factor

NO – must choose another factor NO, must choose another factor 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 1.609 1 # m = 27 s X 1000 m 60 1 60 s X Do units match? NO – must choose another factor Do units match? Cross out common factors NO, must choose another factor

NO, must choose another factor NO – must choose another factor 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 1.609 1 # m = 27 s X 1000 m 60 1 60 s X Do units match? YES ! Cross out common factors Do units match? NO, must choose another factor NO – must choose another factor

Factor label example = 12.0675 m/s Do the math 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 1.609 1 # m = 27 s X 1000 m 60 1 60 s X = 12.0675 m/s Do the math = 12 m/s (correct sig fig)

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

Working with metric/SI quantity base unit length meter mass gram volume liter

SI Base Units Base Quantity Name Symbol Length meter m Mass kilogram kg Time seconds s Electric current ampere A Thermodynamic temperature Kelvin K Amount of substance mole mol Luminous intensity candela cd

Working with metric/SI Base Unit gram meter liter m

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

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

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

Conversions with metric/SI Example: How many cm are in 1.3 m? # cm = 1.3 m x 1 cm 0.01 m = 1.3 x 102 cm

Conversions with metric/SI When given a problem with 2 “prefixes” always go from prefix to base/base to prefix. Example: converting cm to km – convert cm (prefix) to meters (base), then meters (base) to km (prefix). These are often referred to as “2 step problems”

Conversions with metric/SI Example: How many km are in 2.7 x 104mm? This would be considered a “2-step problem”. There are 2 prefixes – km to mm

Conversions with metric/SI Example: How many km are in 2.7 x 104mm? # km = 2.7 x 104mm 1 mm x 0.001 m Base Unit gram meter liter Use chart to get conversion

Conversions with metric/SI Example: How many km are in 2.7 x 104mm? # km = 2.7 x 104mm Use chart to get conversion 1 mm x 0.001 m x 1 km 1000 m Base Unit gram meter liter

Conversions with metric/SI Example: How many km are in 2.7 x 104mm? # km = 2.7 x 104mm 1 mm x 0.001 m x 1 km 1000 m = 0.027 km

Take time now to work on the practice problems Ask questions if you need help!!