Apologia Advanced Physics Sample

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Apologia Advanced Physics in Creation Sample

(All figures, examples & equations are written out in the book but they are not available for this web sample.)

Module #1: Units and Vectors Revisited

Introduction
There are probably no concepts more important in physics than the two listed in the title of this module. In your first-year physics course, I am sure that you learned quite a lot about both of these concepts. You certainly did not learn everything, however. Whether we are talking about units or vectors, there is simply too much information to possibly learn in just one year. As a result, we will take another look at both of these concepts in this first module. This will help you "warm up" to the task of recalling all of the things you learned in your first-year physics course, and it will help to learn both of these valuable concepts at a much deeper level.

Units Revisited
Almost regardless of the physics course, units should be covered first, because a great deal of physics is based on properly analyzing units. In your first-year course, you were taught how to solve problems such as the one in the following example:

EXAMPLE 1.1
A sample of iron has a mass of 254.1 mg. How many kg is that? In this problem, we are asked to convert from milligrams to kilograms. We cannot do this directly, because we have no relationship between mg and kg. However, we do know that a milligram is the same thing as 0.001 grams and that a kilogram is the same thing as 1,000 grams. Thus, we can convert mg into g, and then convert g into kilograms. To save space, we can do that all on one line:

Did this example help dust the cobwebs out of your mind when it comes to units? It should all be review for you. I converted the units using the factor-label method. Because this is a conversion, I had to have the same number of significant figures as I had in the beginning, and even though it was not necessary, I reported the answer in scientific notation. If you are having trouble remembering these techniques, then go back to your first-year physics book and review them.

There are a couple of additional things I want you to learn about units. I am not going to show you any new techniques; I am just going to show you new ways of applying the techniques that you should already know. Consider, for example, the unit for speed. The standard unit for speed is

However, any distance unit over any time unit is a legitimate unit for speed. Since that is the case, we should be able to convert from one unit for speed to any other unit for speed. Study the following example to see what I mean.

EXAMPLE 1.2
As of 2001, the record for the fastest lap at the Indianapolis 500 ("The greatest Spectacle in Racing") was held by Arie Luyendyk. He averaged a speed of 225.2 miles per hour over the entire 2.5-mile stretch of the Indianapolis speedway. What is that speed in meters per second?

This problem requires us to make two conversions. To get from miles per hour to meters per second, we must convert miles to meters. Then, we must convert hours to seconds. This is actually easy to do. Remember, in miles per hour, the unit "miles" is in the numerator of the fraction and the unit "hours" is in the denominator. Also remember that there are 1609 meters in a mile and that 1 hour is the same as 3600 seconds.

Although there is nothing new here, you probably haven't seen a conversion done in this way. Despite the fact that the unit for speed is a derived unit, I can still do conversions on it. I could have just converted miles to meters and gotten the unit meters/hour. I also could have just converted hours to seconds and gotten miles/second. In this case, however, I did both. That way, I ended up with meters/second. When working with derived units, remember that you can convert any or all units that make up the derived unit. Thus, 225.2 miles per hour is the same thing as 100.7 meters per second. Please note that although 3600 has only 2 significant figures, the number is actually infinitely precise, because there are exactly 3600 seconds in an hour. Thus, it really has an infinite number of significant figures. This is why I say that the best rule of thumb is to always end your conversion with the same
number of significant figures as that with which you started your conversion.

Okay, we are almost done reviewing units. There is just one more thing that you need to remember. Sometimes, units have exponents in them. You were probably taught how to deal with this fact in your first-year physics course, but we need to review it so that you really know how to deal with it.

EXAMPLE 1.3
One commonly used unit for volume is the cubic meter. After all, length is measured in meters, and volume is length times width times height. The more familiar unit, however, is cubic centimeters (cc) which is often used in medicine. If a doctor administers 512 cc of medicine to a patient, how many cubic meters is that? Once again, this is a simple conversion. If, however, you do not think as you go through it, you can mess yourself up. We need to convert cubic centimeters to cubic meters. Now remember, a cubic centimeter is just a cm3 and a cubic meter is just a m3. We have no relationship between these units, but we do know that 1 cm = 0.01 m. That's all we need to know, as long as we think about it. Right now, I have the following relationship: 1 cm = 0.01 m This is an equation. I am allowed to do something to one side of the equation as long as I do the exact same thing to the other side of the equation. Okay, then, let's cube both sides of the equation:

Now look what we have. We have a relationship between cm3 and m3, exactly what we need to do our conversion!

So 512 cc's is the same as 5.12 x 10-4 m3.

When most students do a conversion like the one in the example without thinking, they simply use the relationship between cm and m to do the conversion. That, of course does not work, because the cm3 unit does not cancel out, and you certainly don't get the m3 unit in the end:

Do you see what happened? The cm unit canceled one of the cm out of cm3, but that still left cm2. Also, since m is the unit that survives from the conversion relationship, you get the weird unit of m×cm2! When you are working with units that have exponents in them, you need to be very careful about how you convert them. At the risk of "beating this to death," I want to combine the previous two examples into one more example.

EXAMPLE 1.4
The maximum acceleration of a certain car is 21,600 miles per hour2. What is the acceleration in feet per second2? Once again, this is a derived unit, but that should not bother you. All we have to do is convert miles into feet and hours2 into seconds2. There are 5280 feet in a mile, so that conversion will be easy. We do not know a conversion between hours2 and seconds2, but we do know that: 1 hour = 3600 seconds To get the conversion relationship between hours2 and seconds2, then, we just square both sides: (1 hour)2 = (3600 seconds)2 1 hour2 = 1.296 x 107 seconds2 Once again, please note that the conversion relationship between hours and seconds is exact. Thus,
both numbers have an infinite number of significant figures. That's why I reported all digits when I squared 3600 seconds. Now that I have the conversion relationships that I need, the conversion is a snap:

Notice once again that had I not squared the conversion relationship between hours and seconds, the units would not have worked out. In order for hr2 to cancel, the unit hr2 had to be in the problem. That's why it is important to watch the units and make sure they cancel properly.