Report

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 = 12 __ inches 1 kilometer = ____ 1000 meters 1 inch = 2.54 centimeters 4 quarts 1 gallon = __ 1 acre = 4840 square yards 24 hours 1 day = ___ Conversion factors…cont. Conversion factors for 1 ft = 12 in 1 foot 12 inches or 12 inches 1 foot There are almost an infinite number of conversion factors that include meters: 1000 m 1m 1m , , 1 km 100 cm 1000 mm 1m 1m 0.9144 yards , , 3.28 feet 39.37 inches 1m 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 = 4 quarts 1 • 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 answerwhy? • 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) Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km First write down the desired 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 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 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 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 Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 mi x 1 km 0.621 mi Cross out common factors Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 Cross out common factors Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 Are the units now correct? Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 Yes. Both sides have km as units. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 Yes. Both sides have km as units. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 = 75.7 km Now finish the math. Factor label example Q - How many kilometers are in 47 miles? (note: 1 km = 0.621 miles) # km = 47 x 1 km 0.621 = 75.7 km 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. 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 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 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 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 1.609 km 1 mi 1 mi 1.609 km 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 # m = 27 mi s 1 hr 1.609 km X 1 mi 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 s 1 hr 1.609 km X 1 mi 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 s 1 hr 1.609 km X 1 NO, both sides Are the units now correct? 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 s 1 hr 1.609 km X1000 m X 1 1 km Must Cross choose out common another factors 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 s 1 hr 1.609 X 1 X1000 m 1 NO, must choose Do units match? 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 s 1 hr 1.609 X 1 X1000 1 m X 1 hr 60 min Cross out common Do units match 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= s 27 1 1.609 X 1 X1000 1 m X 1 60 min NO Cross NO,–must must out common choose choose Do units match? another factors 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= s 27 1 1.609 X 1 1000 m 1 X X X 1 min 1 60 min 60 s NO Cross NO,–must must out common choose choose Do units match? another factors 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= s 27 1 1.609 X 1 1000 m 1 X X 1 60 X 1 60 s NO Cross NO,–must must out common choose choose Dounits units match? Do match? another factors 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= s 27 1 1.609 X 1 1000 m 1 X X 1 60 X 1 60 s NO Cross NO,–must must out common choose choose Dounits units match? Do match? another factors factor YES ! 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 1 1.609 X 1 = 12.0675 m/s = 12 m/s (correct sig fig) 1000 m 1 X X 1 60 X 1 60 s Do the math • 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 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 Length Base Unit gram meter liter Working with metric/SI 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 x 1000 m = 1.2 x 104 m 1 km 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 = 1.3 x 102 cm 0.01 m 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 x 0.001 m 1 mm Base Unit gram meter liter Use chart to get conversion Conversions with metric/SI • Example: How many km are in 2.7 x 104mm? Use chart get # km = 2.7 x 104mm x 0.001 m x 1tokm 1 mm conversion 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 x 0.001 m x 1 km = 1 mm 1000 m 0.027 km • Take time now to work on the practice problems • Ask questions if you need help!!