Unit 3 Factor Label (Dimensional Analysis)
Metric Prefixes Prefix Symbol Meaning Factor Scientific Not kilo k 1000 times larger than the unit 1000 10 3 deci d 10 times smaller than the unit 1/10 10-1 centi c 100 times smallerthan the unit 1/100 10-2 milli m 1000 times smaller thanthe unit 1/1000 10-3 Metric Units of Length Unit Symbol Relationship Example kilometer km 1 km = 1000 m 5 city blocks (Dairy Queen to JE Jamerson) meter m base unit height of doorknob from the floor decimeter dm 1 m = 10 dm diameter of naval orange centimeter cm 1 m = 100 cm diameter of a shirt button millimeter mm 1 m = 1000 mm thickness of a dime
Conversion Factors Conversion factors are ratios that are equal to one. For example, 1 foot = 12 inches Put it in the form of a fraction or both fractions are equal to 1 The top and bottom of each fraction represent the same distance.
Factor Label Method A factor-label bar utilizes conversion factors to convert a number with a certain unit to a different unit. A factor-label bar is just a huge fraction. The top is the numerator and the bottom is the denominator. There can be as many sections to the bar depending on how many conversion factors are needed to complete the conversion.
Each section of the bar can hold a conversion factor These conversion factors are placed into the bar so that units that can cancel, will cancel. Each vertical line means to multiply the numbers together that it is separating. The answer to the top of the bar will be divided by the answer to the bottom of the bar.
Let s convert 575 inches to miles There are 5280 feet in one mile and 12 inches in one foot Start the bar with the given (575 inches) 575 inches 1 foot 1 mile = 12 inches 5280 feet (inches cancels inches; foot cancels feet; and you are left in miles) =.0090751 = 9.08 x 10-3 miles
Handy Conversion Factors (you should learn them) Distance Mass Volume 1 km =1000 m 1 kg = 1000 g 1 L = 1000 ml 1 m = 10 dm 1 g = 10dg 1 ml = 1 cm 3 1 m = 100 cm 1 g = 100 cg 1 L = 1 dm 3 1 m = 1000 mm 1 g = 1000 mg 1 dm 3 = 1000 cm 3 Notice the similarity between the distance and the mass columns. The prefixes will have the same relationship no matter what the base unit is.
Convert 174 decades to seconds 174 dec 10 years 1 dec 365 days 1 year 24 hrs 60 min 60 sec 1 day 1 hr 1 min = All units cancel, except for seconds which is what we want. If you are using a scientific calculator, the answer will pop into scientific notation. If you are using one of my calculators, you will be getting an error because the number is too big for the calculator s display. The way to ascertain the answer is to place all numbers into scientific notation and then use the rules for multiplying numbers in scientific notation. (174 x 10 2 )(1 x 10 1 )(3.65 x 10 2 )(2.4 x 10 1 )(6 x 10 1 )(6 x 10 1 ) = 548.7264 x 10 2+1+2+1+1+1 = 548.7264 x 10 8.cut off for sig digs and manipulate scientific notation, the answer becomes 5.49 x 10 10 sec The fancy calculator will give you 5.487264E10.the E10 means x 10 10
Convert 85 mm to dm 85 mm 1 m 1000 mm 10 dm 1 m = 850 1000 =.85 8.5 x 10-1 dm Looking at the list of distances, we know that there are 1000 mm in 1 meter and 10 dmin 1 meter Start the bar with the given (85 mm) Place each conversion factor into the bar so that units cancel Multiply the top; multiply the bottom Divide the top by the bottom Answer must be in scientific notation to the correct number of significant digits (the given determines # of sig digs)
More complex conversions Sometimes you will need to convert numbers that have more complex units. Speed units (distance/time) Density units (mass/volume) Area units (distance squared) Odd volume units (distance cubed)
Scenario 1 speed or density units Convert 75 km/hrto cm/sec; the trick is to split the unit in the bar 75 km hr 1000 m 1 km 100 cm 1 m 1 hr 60 min 1 min 60 sec = 7500000 3600 2083.3333 = 2.1 x 10 3 cm/sec Start the bar with the given.only you must split the unit in the bar. Then decide which you want to get rid of first (either distance or time). I chose to get rid of distance first, then time. All units cancel except for cm (up top) and sec (on the bottom).
Scenario 2 area units or odd volume (not in the volume list) Convert 83 m 3 to cm 3 ; cm 3 is in the list but m 3 is not; the trick is to use distance conversion factors as many times as needed to cancel the unit. 83 m 3 100 cm 1 m 100 cm 1 m 100 cm 1m = 83000000 8.3 x 10 7 cm 3 Start the bar with the given if you look at the volume conversion factors, you will not see one that has m 3 in it. So you need to use the distance conversion factor ( 1 m = 100 cm) three times in order to completely cancel out the m 3 unit. This leaves you with cm x cm x cm which is cm 3.