9-4-96
Relevant sections in the book : 1.1, 1.2, 1.3, and 1.4 (also Appendix A - Appendix E)
If you were taking a trip to Greece, you'd get the most out of your trip if you learned some Greek before you got there. Knowing a little of the language would help you somewhat; being fluent in the language would help you immensely. The same holds true of physics. If you're fluent, or at least knowledgeable, in the language of physics you'll find this course a lot easier than if you don't know any of the language. The language of physics is often math (although, in some cases, Greek is involved too), such as trigonometry and algebra. There are other things, such as the metric system, that provide a common reference frame.
This introduction to physics is simply a review of the language of physics. If you're unfamiliar with some of these concepts, spending the time to become fluent will help you immensely in the course. Practice makes perfect!
If you and your friend are discussing the 200-meter world-record holder, Michael Johnson, and you know his speed in miles/hour while your friend wants to use meters/second, you may have some trouble understanding each other. This happens in physics, too, so it helps to have a common system of units, just to make life easier. In physics, the MKS system of units is generally used, where M stands for meter, K for kilogram, and S for seconds. These are the units of length, mass, and time. There are often cases where another system of units might be easier to work with, but for now let's just worry about one.
MKS units are part of the metric system, based on powers of ten to keep things simple. The more common prefixes used in the metric system, and the power of ten associated with them, are given in the table below. You should know all of these off by heart.
A more complete table is found on page 5 in the textbook.
You will often need to convert a value from one unit to another, converting from meters to centimeters, for example. How many centimeters are in 200 meters? Simply multiply by 100 cm / 1m (the same distance, in different units) to cancel the units of m and leave you with cm.
200 m x (100 cm /1 m) = 20000 cm or 2.00 x 10^5 cm.
Be extra careful if you have units which are squared, cubed, or have some other exponent. If you have a small box which is 10 cm on each side, then the volume is simply the length x the width x the height:
V = 10 cm x 10 cm x 10 cm = 1000 cm^3.
In m^3, the volume is
Let's try a slightly more complicated example, with two steps rather than one. When he set the 200-meter world record, 19.32 seconds, Michael Johnson ran at an average speed of 200 m / 19.32 s = 10.35 m/s. If you want to know how fast this is in miles/hour, the conversion would be carried out like this:
10.35 m/s x (1 mile / 1609 m) x (3600 s / 1 hour) = 23.2 miles / hour.
All you do to convert is to multiply the original value by x/y, where x and y are the same thing, expressed in different units. Here we did that twice. There are 3600 seconds in 1 hour, for example, and 1609 meters in 1 mile. To figure out which goes on top and which on the bottom, figure out where you need to get cancellation of units. We wanted to convert from meters to miles in the example above. The m was on top, so we had to put the 1609 m on the bottom so the two units of meters would cancel, leaving miles on top.
Something else to keep in mind is that when values are multiplied or divided, they can have different units. When you add or subtract values, however, the values must have the same units.
When you punch in 200/19.32 in your calculator, your calculator gives you the number 10.35196687. You have to round this off (10.35 is a good choice), because most of the figures the calculator gives you are not significant, meaning, essentially, that they're meaningless. The calculator assumes that what you type in is accurate to about 12 figures; typically, your numbers are accurate to 2 or 3 or 4 figures. The Olympic timing system, for example, gave Michael Johnson's time as 19.32 seconds, which means that his time was 19.32 +/- 0.005 seconds. It can't be given with infinite precision, so it gets rounded off to the nearest hundredth of a second, giving 4 significant figures. The distance that Michael Johnson ran, 200 meters, is probably measured to the nearest centimeter, so it's accurate to about 5 significant figures (200.00 +/- 0.005 m).
When you're combining numbers, adding, subtracting, multiplying, dividing, or whatever, the general rule of thumb is that you should round off your answers to the same number of significant figures as there are in the number with the smallest number of significant figures. In the example above, dividing two numbers with 4 and 5 significant figures, round off to 4.
Another general rule of thumb : if you have to calculate intermediate values before you get to a final value, keep one or two extra significant figures for the intermediate values, just so you won't introduce any inaccuracies in the final answer by rounding off too soon.
Basic trigonometry is usually introduced by looking at a right-angled triangle. Let's use a triangle with sides of length 3 cm, 4 cm, and 5 cm. This satisfies the Pythagorean theorem, which states that for a right-angled triangle with sides a and b, and with a hypotenuse c (the side opposite the 90-degree angle),
In our example, 3^2 + 4^2 = 5^2.
Let's say we'd like to figure out the angle between the 3 cm side and the 5 cm side. We'll call this angle by the Greek letter theta. For a right-angled triangle, we can use the following relationships:
where hyp stands for hypotenuse, opp stands for the side opposite to theta, and adj stands for the side adjacent to theta (the adjacent side that's not the hypotenuse). In the case of our 3-4-5 triangle, we have
Note that the units cancel out, so these values have no units. All these relationships give the same answer for theta, 53.1 degrees: inverse sin of 4/5, inverse cos of 3/5, and inverse tan of 4/3.
If we wanted to find the third angle in the triangle, we could use geometry, remembering that the three angles in a triangle must add up to 180 degrees. We could also use trigonometry, using, for example, inverse sin of 3/5 to get 36.9 degrees.
The sine relationship used above for the right-angled triangle, as well as the Pythagorean theorem, are simply special cases of the sine law and the cosine law, respectively. Most triangles do not have 90-degree angles in them, but there are still rules we can use to relate sides and angles. In any triangle, with sides a, b, and c, the sine and cosine laws are as follows:
Be extra careful with the sine law. If the angle is greater than 90 degrees, your calculator will give you 180 - the angle, instead of the angle (your calculator will always give you an angle less than 90 degrees). To find the angle, subtract what your calculator gives you from 180 degrees.
Algebra basically involves the manipulation of equations to solve for an unknown variable. PY105 involves a great deal of problem solving, which requires a lot of reasonably-straightforward algebra. An example will help illustrate the sort of manipulations you'll be expected to carry out. Let's say you have an equation that states:
and you're given a = -2.00 m/s^2 and v = 2.00 m/s. You don't know x, so you have to rearrange the equation to solve for x, which means getting x by itself on the left side. Note that the equation involves numbers, variables, and units...it would be easy to confuse a unit with a variable, but usually it's fairly obvious what is what. In this case, v, a, and x are variables, while m and s are units (meters and seconds).
First of all, let's analyze the units. The left hand side has units of m^2/s^2, and the first term on the right does also. For the equation to make sense, the second term on the right must have the same units, which means x must have units of m. x is a length, in other words.
Now let's rearrange the equation to solve for x. The units will be left out, because we know what the unit for x is, but in some cases it's wise to keep track of the units as you do a calculation. You can do several steps at once, but let's just do one at a time:
x = (4.00 -25.00) / (-4.00) = -21.00 / -4.00 = 5.25 m
Just one word of caution about relying too much on your calculator. It's very easy to make a mistake in what you plug in to your calculator; just because your calculator gives you an answer doesn't mean it's the correct answer. Think about whether your answer makes sense. Is it about the right size, or is it much too large or too small? Does the answer have the right sign? Does the answer make physical sense? It's always a good idea to do a "back-of-the-envelope" calculation to come up with an approximate value for the answer...round your numbers off to easily-calculated values, work out a rough answer on paper or in your head, and then plug the exact values into your calculator to get the answer, making sure the answer is in the same ballpark as your rough answer.
Doing back-of-the-envelope calculations will help you get a feel for numbers, and more of a feel for physics. Relying solely on your calculator will often get you the right answer, but if you go wrong you'll be less likely to realize it.