9-3-99
Sections 1.1 - 1.8 and Appendix A
If you were taking a trip to Greece, you'd get the most out of your
trip if you learned some Greek before going. Knowing a little of the
language would help you somewhat; being fluent in the language would
help you immensely. The same holds true for physics.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!
MKS units and the metric system
MKS (meter-kilogram-second) 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 powers of ten associated with them, are
given in the table below. You should know all of these off by heart.
Unit analysis and conversion
You will often need to convert a value from one unit to another,
converting from centimeters to meters, for example. Be extra careful if
you have units which are squared, cubed, or have some other exponent.
If you have a cube 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 cm3.
In m3, the volume is
Consider 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 multiply the original value by x/y, where x
and y are the same thing, expressed in different units. 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.
Significant figures
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.
When you're combining numbers, 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.
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.
Trigonometry
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¡ angle), the square of the hypotenuse equals the
sum of the squares of the other two sides.
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 ,
and adj stands for the side adjacent to (the adjacent side that's not
the hypotenuse). Note that the units cancel out, so these values have
no units.
If we wanted to find the third angle in the triangle, we could use
trigonometry, or geometry, remembering that the three angles in a
triangle must add up to 180¡.
The Sine law and the Cosine law
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¡ angles
in them, but there are still ways to relate sides and angles. In any
triangle the sine and cosine laws are as follows:
Be extra careful with the sine law. If the angle is greater than 90¡,
your calculator will give you 180 - the angle, because your calculator
gives you an angle less than 90¡. To find the angle, subtract what your
calculator gives you from 180¡.
Algebra
Algebra involves the manipulation of equations to solve for unknown
variables. PY105 involves a great deal of problem solving, requiring 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/s2 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 m2/s2,
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.
You don't have to do this in one step, but you should be able to re-arrange the equation above to solve for x:
Getting a feel for numbers
Just one word of caution about relying too much on 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.
