Conversion Factors and Dimensional Analysis

[narrator] Let s talk about dimensional analysis
and conversion factors. Dimensional analysis is going to use the units from a given quantity,
or measurement, just from the problem–it will be given to us, and it will use a conversion
to extract useful information or to swap conventions. Say from the SI system or metric system to
US or back and forth. We can swap conventions or we can extract useful information, for
example if I’m given a density of .323 grams per milliliter and I want to know, so this
is given–and I want to know what is the mass then I know that I ve got to multiply the
units by milliliters so that they cancel out, and I’m left with grams. I don’t need to know
the formula of density equals mass over volume if I’m given the units because I can figure
it out using dimensional analysis. What I need to do to get what I’m going for. What
am I desiring in the solution to the problem? I can use those units in a conversion factor.
What are conversion factors? Well, conversion factors start with us being given a unit.
The conversion factor itself, that I’m going to surround with this blue box here, is going
to have a new unit in the denominator which is exactly equivalent to the old unit, the
given unit, in the denominator. The whole point is something divided by itself is equal
to one. When I multiply and divide I’m left with a number, a new number, and a new unit.
The whole point is a conversion factor. We’re assuming, for example, here s one that we
need to learn for the course. We know that 2.54 centimeters is equal to 1.00 inches,
to three significant figures. I can make a conversion factor since these two quantities
are equal. I can divide both sides by either quantity. I could choose to divide both sides
by centimeters, 2.54 centimeters would give me 1 equals 1.00 inch over 2.54 centimeters,
or I could go the other way and I could say divide both sides by inches, by 1.00 inch,
and I would get 1 equals 2.54 divided by 1.00 inch. The point is if I multiply this whole
quantity equals one. Since it equals one I m not changing anything about it. I’m just
converting from one unit to the other. Let s see this in practice. Let s do an example.
How many kilometers is the speed of light? We’re given that the speed of light is 3.00
x 108 meters per second. What do I want? I want kilometers per hour, not meters per second.
I’m starting with a unit and I want to get rid of that unit. It has to go on the bottom.
What do I want to convert to? Well, I want to go from meters to kilometers. I can cancel
the meter units but I need to know a conversion factor. What equals what? Well, that k on
kilometer means 1,000. I can substitute 1,000. There s 1,000 meters in 1 kilometer. I learned
that from my prefix table. There s a good conversion factor. This one works. There’s
our first conversion. Then we keep going. Well, I’ve gotten one part of this. I’m there.
The next part I want to get to hours. Well, I don t have hours. I can go directly to hours
if I know how many seconds are in an hour but let s just say let s get rid of the unit
we don’t want. I put it on top so that it divides second by second to give 1. Then I
can go to minutes. I know how man seconds are in a minute. There s 60 seconds in 1 minute.
These quantities are equivalent, conversion factor one, conversion factor two, I can keep
going. Well, now I’m rid of seconds but I don t have hours, yet. I need one more conversion
factor. I want to get rid of minutes so I put minutes on top and I want to get hours.
60 minutes in 1 hour. Minutes goes away and when I simplify down the only things that
don t cancel; kilometers on top, hours on bottom. Now it s just a matter of multiplying
it out. In our calculator, one thing I wanted to point out that I didn’t before is that
we can punch this in, usually by saying 3.00 E, means 10 to the–means times 10 to the,
and then positive 8. That s one way we can put that in the calculator. Make sure we re
practicing that. Let s punch it out. 3 x 108 divided by 1,000, times 60, times 60. I should
108–one, two, three, four, five, six, seven zeroes. Really big number. We can put that
in science notation. Here s our decimal place. One, two, three, four, five, six, seven, eight,
nine. 1.08 x 109 kilometers per hour.

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