Dimensional Analysis

>>Hi everyone, and welcome
to the Penguin Prof Channel. Today I’m going to continue
my College Success Series with a math skills review
on dimensional analysis. So first of all what is it? Some people call this
the unit factor method. It goes by several
different names. But it actually is an
essential concept that you use in all areas of life,
really, not just science. Especially if you travel
you’ve run into this. Because you travel to
different countries that use different
currencies and different units. And you are probably
doing dimensional analysis without even realizing it. So, for example, if you travel
to one of my favorite countries, like Italy, and you do some
shopping or you do some eating, you find yourself
wanting to buy things. And wondering, well you know, how much does this
cost in US dollars? And in order to answer
that question, you’re doing dimensional
analysis. Dimensional analysis is based
on one really simple idea. Which is, that you can multiply
anything by the number 1 without changing its value. And you already know that. You can multiply anything by 1, and you get anything [laughs]
whatever that number was. No matter how big and ugly
the number, you multiply it by 1, it doesn’t change. So you’re allowed to do that. And that’s it. Now here’s the thing, you’re
used to looking at the number 1 and seeing it a certain way. But the number 1 can look
very, very different. So these are other
examples of the number 1. 2 divided by 2. 14.5 divided by 14.5. Even a penguin divided
by a penguin [laughs] — you know I had to
get that in there. Those are all equivalent
to the number 1. When you do currency
conversions and you have to look up what the currency conversion
is for a particular country that you’re going
to be travelling to. I just looked this up. We’re going to be travelling
to Peru in a couple weeks. And about 11 US dollars will
give you about 30 Nuevo soles. This is important because
that allows you to travel to countries where they have,
for example, humble penguins. So you’re going to be running
into conversions a lot. There’s a lot of other
ways to be the number 1. A dozen eggs is the
same thing as 12 eggs. So you can put those two values and set them equal
to each other. There’s 60 minutes in an hour. So you know that
that’s equivalent. In one year there’s 365 days. These fractions are all
equivalent to the number 1. And it doesn’t matter
who’s on top. So you can put — for the egg
example, you can put the 12 eggs in the numerator or the
denominator and match that up with a dozen eggs. It’s the same thing. You can put the 60
minutes divided by an hour. Or you can say 1 hour
divided by 60 minutes. Does not matter. Same thing with the
year and the days. So it’s really, really useful
to understand that no matter where you put these
numbers, you can see them as an equivalent so
that it equals 1. One of the most important
things though is that you keep track
of the units. Don’t drop the units,
because the numbers by themselves don’t
mean anything. So let’s see the power
of number 1 in action. How many centimeters
are there in 6 inches? You have to know the
conversion there. 1 inch=2.54 centimeters. So how you’re going to do this
is — this is my suggestion, is that you set up each problem
by writing down what you need to find with a question mark. And I like that because the
question mark shows you that’s what the question is. You’re going to set that equal to the information
that you’re given. And then solve the problem by multiplying the given
data and its units. This is really important
here — its units — by the appropriate unit factors. So that only the units
that you want are left. That means you’re
going to cancel out all the other units
that you don’t want. If you lose the units, you’re
going to lose the problem. You must keep the
units in there. So this question I’m going to
start with how many centimeters. Because that is what
I’m being asked. How many centimeters
are there in 6 inches? So I’m going to set
that equal to the units that I want in inches. I’m going to multiply that
by the conversion factor. 2.54 centimeters per inch. And I choose the
centimeters divided by inches, because I want inches
to cancel out. In other words, when inches
are in the denominator, and also here — this is
a numerator, remember? I can cancel them out. That’s going to leave me
with the units I want. That leaves me with centimeters. So if I do that math. And I do that 6 times 2.54, and
it’s actually divided by 1 — which doesn’t matter —
I get 15.2 centimeters. So that means that in 6 inches, that 6 inches is equivalent
to 15.2 centimeters. And that’s going to three
significant figures. Here’s another example. Express 24 centimeters
in inches. So you’re going to
do the same idea, except that this time my
desired units are different. I want inches. So I’m going to set
it up differently. So I’m going to use? inches=24 centimeters. And I’m going to use
the same conversion, but notice how I
choose to put inches over centimeters in this case. That’s because I want
centimeters to cancel out. So I have centimeters
in the denominator here. I have centimeters in the
numerator here, that realize that a whole number,
that is a numerator. Centimeters cancel and I
end up with 9.45 inches. So hopefully you’re kind of
getting the hang of that. You can use multiple
unit factors at once. And we’re going to
do this example. How many seconds are
there in two years. So I’m going to start
with what I want, how many seconds
in equals 2 years. Now I’m going to have to
somehow get rid of years and end up with seconds. And I’m going to do
that by multiplying by several different
ways of saying 1. I’m going to get rid
of years by multiplying by 365 days divided by a year. That’s 1. I’ve got to get
rid of years now by — or excuse me, days — by multiplying 24 hours
in a day — that’s 1. I’m going to get rid of hours by multiplying how many hours
there are in a minute — 60 minutes divided by an hour. And finally I’m going
to get to seconds by converting how
many seconds are there in a minute, which is 60. So if you do all the math
— and this is a big number. 2 times 365 times 24
times 60 time 60 — I didn’t need to
divide by anything, because all the denominators
here are 1. I get 63,072,000
seconds in two years. And if your instructor is like
me, you really need to convert that to scientific notation. So I’m going to make this 6.3 — I’m going to put a little
decimal point there. And I need to count how
many places I’m going to be moving this decimal. So this is going to be from 6.3, I’m going to be moving
it 1, 2, 3, 4, 5, 6, 7. So that’s going to be 6.3
times 10 to the 7th seconds. And again, I’m using
the same number of significant figures
as is appropriate. We can do another module
on significant figures at a later time if you want. So that’s the idea. And here you can use multiple
unit factors in order to get to the answer that you want. The key is going to be
hold onto those units. Because if you find yourself
multiplying by, you know, for example, 365 divided by 1,
and you don’t put the units in, that’s actually not 1 is it? That number would be 365. So if you kill the units,
you’re going to kill it. I don’t give any credit if
my students drop the units. The units are everything. So some things to know. It can be really helpful
to get comfortable with prefixes in
the metric system. Most of us in science,
we focus on kind of one side or the other. Usually in biology —
or in physiology anyway, we focus on kind of
the lower part of this. Most of you will probably
be focused somewhere here. These are just helpful in
knowing how many places over, you know, you move from
the basic unit of 1. So this would be, for example,
you know, 1 meter or 1 liter. Then you can change
your prefixes, and then very easily
indicate how many powers of 10 you’re talking about. So that’s really useful to know. It’s also useful to
have some references. Especially conversion
tables like these. You want to have
access to these. All of these conversions
are different ways of saying 1, right? So if you have a question, like how many kilograms does a
90-pound Emperor penguin weigh? You’re going to look at the
section of the table under mass. And you’re going to use the
conversions where you see pound, and then you see kilograms. So if you use 0.45 kilograms per
pound, that’s going to allow you to answer this question. How many kilograms does a
90-pound Emperor penguin weigh? This is something that of
course, everyone needs to know. Now I want you to be careful. I know that online you can get
all kinds of widgets and apps which will do the
conversions for you. And that’s great if you
happen to have access to them. But I do require my
students to be able to use the conversion
factors that I give them on an exam to do it themselves. So just be aware that you do
need to know how to do this. Once you master it, of
course, then you know, it’s easy to plug in values. If you have the power of
conversions, you’ll be able to travel and enjoy your life. And convert anything you
want into anything else. And buy your pizza. This is my favorite pizza. This is a beautiful medieval
town called Siena, in Tuscany. This is not Photoshopped. I know — people say
this picture looks fake. But it was real. I was just eating my pizza
and put it on the wall there. It was just perfect. And anyway, it was the power
of conversion that allowed me to know how much this pizza
cost in US dollars [laughs]. Oh, it’s hard to make
math sexy isn’t it? Thank you so much for visiting
the Penguin Prof channel. Please rate and subscribe. Visit on Twitter and
the Facebook page. Good luck.

7 thoughts on “Dimensional Analysis

  1. I'd like to hire you to tutor me in everything; how to clean a spoon, the physiology of goosebumps, differential calculus, how to correctly use a semicolon. Unfortunately for us both, I cannot afford the $430/hr rate your individual  instruction is obviously worth.

  2. Would it be possible to get the link to the conversion chart resource you showed in the video (9:18)? I am trying to print a copy of it. Thank you for the videos

  3. you have a voice for narrating like no other. I could not imagine these videos with out your voice. and such a great voice it is. I thought I tell you. and the videos aren't bad either. thank you.

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