[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

factor

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.