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# Appropriate units

Video transcript

Now that we know that we can
treat units algebraically, let's see if we can
use that knowledge to figure out what the
units of a variable might be from some formula. So let's say that I am given
this formula, that capital K is equal to lowercase
b over s squared. And let's say that
we are also told that b is going to
be in centimeters, and s is going to be in
grams per centimeter. So that's the units of s
is grams per centimeter. B is in centimeters. What is the units
of K going to be? And I encourage you
to pause this video and think about this. Well, let's try it out with
just some value and some values for b and s, and
see what happens. Let's say that b is
equal to 1 centimeter, and I'm just using 1, because
it makes the math really easy. And let's say that s is
1 gram per centimeter. So, we know that
K is going to be equal to 1 centimeter-- that's
b-- over 1 gram per centimeter squared. Now, we've already
talked about the idea that we should treat our units
like algebraic quantities, so this should be equal to
1 centimeter over-- now, you could treat this
as 1 squared times-- let me rewrite this. You could rewrite this
as 1 squared times grams per centimeter squared, and
this just comes out of the idea that if I have a
times b to some power, say, the second power, that's
equal to a squared b squared. And we also know that if I
have a/b to some power, say, squared, that's the same thing
as a squared over b squared. So I could rewrite
this as this is going to be equal to 1
centimeter over-- 1 squared is just 1-- 1 gram squared
per centimeter squared. Now, what is this
going to be equal to? Let me write it over here. Well, that's going to be 1
centimeter in our numerator. Let me do it in that
green color, just to keep track of things. 1 centimeter from our numerator,
instead of dividing it by 1 gram squared
centimeter squared, I can multiply it times
the reciprocal of this. So times 1/1, which
is just going to be 1, but let me just
write it that way just so that you get that I'm
taking the reciprocal of 1. So 1/1 times the reciprocal of
grams squared per centimeter squared. Well, that's just
going to be centimeters squared per grams squared. So what's this going
to be equal to? Well, I have 1 times 1, so if I
take the numeric parts, 1 times 1, it's just going to be 1. And now remember, we can
treat the units algebraically. I have a centimeters
times centimeters squared. Well, that's going to get me to
centimeters cubed, centimeters to the third power. And this isn't c times
n to the third power. This is centimeters
to the third power-- or you could write
it like this if you wanted to be clear about it--
per-- and over here we still have grams squared--
per grams squared. So the units, I just picked
the numbers 1 arbitrarily, so the 1 isn't
what matters here. I just picked those numbers. What matters here is
what are the units. What are the units
for this formula when b is in centimeters and s
is in grams per centimeters? We see that K is
going to be expressed in terms of cubic
centimeters per gram squared. And I can't tell
you offhand kind of an intuitive physical
representation of that, but you at least know the units. You have cubic
centimeters, which you can view as a volume--
divided by, I guess, a mass squared. And a mass squared is a little
bit less intuitive for me, but at least we know
what the units of K would be in this formula
given the units of b and s.