One of these things is Not like the other… This guide will explain briefly the concept of units, and the use of a simple technique with a fancy name— "dimensional analysis”. Whatever You Measure, You Have to Use Units • A measurement is a way to describe the world using numbers. We use measurements to answer questions like, how much? How long? How far? • Suppose the label on a ball of string indicates that the length of the string is 150. • Is the length 150 feet, 150 m, or 150 cm? • For a measurement to make sense, it must include a number and a unit. Units and Standards Number 150 feet Unit Rule: No naked numbers. They must have units • Now suppose you and a friend want to make some measurements to find out whether a desk will fit through a doorway. • You have no ruler, so you decide to use your hands as measuring tools. Units and Standards • Even though you both used hands to measure, you didn’t check to see whether your hands were the same width as your friend’s. Units and Standards • In other words, you didn’t use a measurement standard, so you can’t compare the measurements. • Hands are a convenient measuring tool, but using them can lead to misunderstanding. Units and Standards • So in order to avoid confusion we use measurement standards. • A standard is an exact quantity that people agree to use to compare measurements. Units and Standards • In the United States, we commonly use units such as inches, feet, yards, miles, gallons, and pounds. This is known as the English system of measurement. • Most other nations and the scientific community use the metric system - a system of measurement based on multiples of ten. International System of Units • In 1960, an improvement was made to the metric system. This improvement is known as the International System of Units. • This system is abbreviated SI from the French Le Systeme Internationale d’Unites. International System of Units • The standard kilogram is kept in Sèvres, France. • All kilograms used throughout the world must be exactly the same as the kilogram kept in France because it is the standard. International System of Units • Each type of SI measurement has a base unit. • The meter is the base unit of length. International System of Units • Every type of quantity measured in SI has a symbol for that unit. • All other SI units are obtained from these seven units. Review • When we measure something, we always specify what units we are measuring in. • All kinds of units are possible, but in science we use the SI system. • Problem! What if I measure something in inches but I am supposed to give you the answer using SI units? Sometimes You Have to Convert Between Different Units • How many seconds are in a day? • How many inches are in a centimeter? • If you are going 50 miles per hour, how many meters per second are you traveling? • To answer these questions you need to change (convert) from one unit to another. How do you change units? • Whenever you have to convert a physical measurement from one dimensional unit to another, dimensional analysis is the method used. (It is also known as the unit-factor method or the factor-label method) • So what is dimensional analysis? The converting from one unit system to another. If this is all that it is, why make such a fuss about it? Very simple. Wrong units lead to wrong answers. Scientists have thus evolved an entire system of unit conversion. Dimensional Analysis • How does dimensional analysis work? • It will involve some easy math (Multiplication & Division) • In order to perform any conversion, you need a conversion factor. • Conversion factors are made from any two terms that describe the same or equivalent “amounts” of what we are interested in. For example, we know that: 1 inch = 2.54 centimeters 1 dozen = 12 Conversion Factors • So, conversion factors are nothing more than equalities or ratios that equal to each other. In “math-talk” they are equal to one. • In mathematics, the expression to the left of the equal sign is equal to the expression to the right. They are equal expressions. • For Example 12 inches = 1 foot Written as an “equality” or “ratio” it looks like =1 or =1 Conversion Factors or Hey! These look like fractions! • Conversion Factors look a lot like fractions, but they are not! • The critical thing to note is that the units behave like numbers do when you multiply fractions. That is, the inches (or foot) on top and the inches (or foot) on the bottom can cancel out. Just like in algebra, Yippee!! Example Problem #1 • How many feet are in 60 inches? Solve using dimensional analysis. • All dimensional analysis problems are set up the same way. They follow this same pattern: What units you have x What units you want What units you have The number & units you start with The conversion factor (The equality that looks like a fraction) = What units you want The units you want to end with Example Problem #1 (cont) • You need a conversion factor. Something that will change inches into feet. • Remember 12 inches = 1 foot Written as an “equality” or “ratio” it looks like 60 inches x = 5 feet (Mathematically all you do is: 60 x 1 12 = 5) What units you have x What units you want What units you have = What units you want Example Problem #1 (cont) • The previous problem can also be written to look like this: • 60 inches 1 foot = 5 feet 12 inches • This format is more visually integrated, more bridge like, and is more appropriate for working with factors. In this format, the horizontal bar means “divide,” and the vertical bars mean “multiply”. Dimensional Analysis • The hardest part about dimensional analysis is knowing which conversion factors to use. • Some are obvious, like 12 inches = 1 foot, while others are not. Like how many feet are in a mile. Example Problem #2 • You need to put gas in the car. Let's assume that gasoline costs $3.35 per gallon and you've got a twenty dollar bill. How many gallons of gas can you get with that twenty? Try it! • $ 20.00 1 gallon = 5.97 gallons $ 3.35 (Mathematically all you do is: 20 x 1 3.35 = 5.97) Example Problem #3 • What if you had wanted to know not how many gallons you could get, but how many miles you could drive assuming your car gets 24 miles a gallon? Let's try building from the previous problem. You know you have 5.97 gallons in the tank. Try it! • 5.97 gallons 24 miles = 143.28 miles 1 gallon (Mathematically all you do is: 5.97 x 24 1 = 143.28) Example Problem #3 • There's another way to do the previous two problems. Instead of chopping it up into separate pieces, build it as one problem. Not all problems lend themselves to working them this way but many of them do. It's a nice, elegant way to minimize the number of calculations you have to do. Let's reintroduce the problem. Example Problem #3 (cont) • You have a twenty dollar bill and you need to get gas for your car. If gas is $3.35 a gallon and your car gets 24 miles per gallon, how many miles will you be able to drive your car on twenty dollars? Try it! • $ 20.00 1 gallon 24 miles $ 3.35 1 gallon = 143.28 miles (Mathematically all you do is: 20 x 1 3.35 x 24 1 = 143.28 ) Example Problem #4 • Try this expanded version of the previous problem. • You have a twenty dollar bill and you need to get gas for your car. Gas currently costs $3.35 a gallon and your car averages 24 miles a gallon. If you drive, on average, 7.1 miles a day, how many weeks will you be able to drive on a twenty dollar fill-up? Example Problem #4 (cont) • $ 20.00 1 gallon 24 miles 1 day $ 3.35 1 week 1 gallon 7.1 miles 7 days = 2.88 weeks (Mathematically : 20 x 1 3.35 x 24 1 x 1 7.1 x 1 7 = 2.88 ) Dimensional Analysis • So you can have a simple 1 step problem or a more complex multiple step problem. Either way, the set-up of the problem never changes. • You can even do problems where you don’t even understand what the units are or what they mean. Try the next problem. Example Problem #5 • If Peter Piper picked 83 pecks of pickled peppers, how many barrels is this? • Peter Piper picked a peck of pickled peppers... Or so the rhyme goes. (What in the world is a peck?) • You need help for this one. As long as you have information (conversion factors) you can solve this ridiculous problem. Example Problem #5 (cont) • Use this info: A peck is 8 dry quarts: a bushel is 4 pecks or 32 dry quarts; a barrel is 105 dry quarts. • WHAT?! Rewrite them as conversion factors if the info is not given to you that way. 8 dry quarts 1 peck or 1 peck . 8 dry quarts 32 dry quarts 1 bushel 105 dry quarts 1 barrel 4 pecks 1 bushel or or or 1 bushel . 32 dry quarts 1 barrel . 105 dry quarts 1 bushel 4 pecks Example Problem #5 • Pick the conversion factors that will help get to the answer. 83 pecks 1 barrel. Hint: Look for units that will cancel each other. • 83 pecks 8 dry quarts 1 peck 1 barrel 105 dry quarts = 6.3 barrels (Mathematically : 83 x 8 1 x 1 105 = 6.3 ) Review • Dimensional Analysis (DA) is a method used to convert from one unit system to another. In other words a math problem. • Dimensional Analysis uses Conversion factors . Two terms that describe the same or equivalent “amounts” of what we are interested in. • All DA problems are set the same way. Which makes it nice because you can do problems where you don’t even understand what the units are or what they mean.