# Measurements

## How do we choose units

The important thing to remember here, is that units are usually chosen to be convenient. I made a couple of jokes about needing a unit for distance that is about the width of your thumb or a unit of volume that’s the right amount to order for a glass of beer (or orange juice, in your case). So, using the units you are more used to, a “cup” of orange juice or a “pint” of beer may seem about right. Measuring out gasoline in cups would be a little less convenient than gallons. Similarly, measuring distances between your home and downtown San Diego in inches would be inconvenient. Thus, we try to have units such that the amount we are measuring provides us with a number that makes sense in our feeble brains. (No offense intended, no humans are any good at comprehending really large or really small numbers.) The problem with the system that you grow up with is that there is no consistent relationship between units that are optimized for measuring out a cup of orange juice, a gallon of gasoline, or a barrel of oil, just as there is no consistent relationship between an inch, a foot, a yard and a mile. You have to remember all of the unit conversions. At this point you probably know most of them however, I have to say it is a bit of pain in the butt. We will be using the SI Units. That stands for the “international system” of units (or rather, it stands for the French version of that). The great advantage here is that in each case we have one base unit, say, the meter for distance, and then units derived from that as factors of 1,000 larger or smaller. In some cases, factors of 10 or 100 may be used if we have need for a unit in that range. A meter is about the length from the tip of your nose to the tip of your finger, that is, if you are an average male in the 17th century (actually, the definition of a meter has been updated many times over the last couple of hundred years). The common unit “centimeter” is one 100th of a meter. Normally, as I said, we would multiply or divide by factors of 1,000. So we have: a millimeter, 1/1000 of a meter; a micrometer, 1/1000 of a millimeter; a nanometer, 1/1000 of a micrometer, and so on. Whether we are speaking about meters, grams, liters, or volts, or any other basic unit, milli means 10
-3, micro means 10-6 and nano means 10-9. The reason for the centimeter is that the length is just convenient for many things we measure. Sure, I could refer to the width of my index finger as about .01 m, or 10 mm, but one centimeter just seems easier (note that all of those values are the same). The same basic strategy applies when we go to larger and larger amounts. I could talk about my new hard drive and the fact that it can store up to 1 trillion bytes, but nobody really comprehends what one trillion means (just try talking about our national debt some time). However if I say 1000 GB, or 1 TB, those are numbers we can think about at least reasonably well. You have a sense from working with storage on your computer about how much a gigabyte really is. I could tell you that something has a mass of 10,000 grams; but, even if you have a sense of what a gram is, you cannot really conceive of what 10,000 of them would be. On the other hand, 10 Kg makes some sense, if you have an idea of what a Kg feels like. Below is a table of the prefixes that we will use commonly. Of the ones listed below, I expect you to know: centi; milli; micro; nano; Kilo and all the abbreviations. The others will rarely come up.
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## Dimensional analysis

The basic idea is this: the units must cancel out to give you the proper units for your answer. If you want your answer to be in grams, the other units in the calculation must cancel. The nice thing about this is that it shows you how to set of the calculation. Rather than try to remember whether two numbers should be multiplied, or one divided into the other, you set up the equation just so that the units cancel. You can set up a long running calculation with the numbers left out, just with units, so that they cancel properly. The you just fill in the numbers. So, here is an example: A car is traveling 60.0 miles/hour. Given that a mile=1,609.344 meters, how many seconds will it take to travel 10,000.0 meters. So, you want the answer in seconds, and the only starting number you have is in miles/hour. You are given the conversion factor for meters/mile, and the others you should know. I'm going to start by inverting miles/hour, because I want time to be on top in the final answer. You could save that step to the last, if you want. you see how miles, hours, meters and minutes cancel leaving seconds? Now, just plug in the numbers where they belong. By the way, am I Okay on the sig. figs? Let's see, the speed is given in 3 sig. figs...so my answer should not have any more than that. What about the others? The 60 minutes in an hour and 60 seconds in a minute are both exact numbers. That is, they are not measurements; they are definitions. The other numbers are more sig. figs. So, I should report it at only three sig. figs, or 373 seconds. Here is another one: A furlong is 220.0 yards and an inch is 2.54 cm exactly. A really fast (but believable) racehorse can run 10 furlongs in about 2 minutes. If a horse wins a 10 furlong race with a time of 2 minutes flat (2.00 minutes), how fast was he traveling in meters/second?