Let's take an example, suppose we have one atmosphere
so we're at normal atmosphere pressure and we want to make a barometer,
again but this time we would like to make a water barometer
and see if this water barometer would work
instead of a mercury barometer since maybe mercury is hard to get our hands on.
So if you have mercury in a barometer normally at atmosphere pressure
at height of 760 millimeters and we try to make a barometer instead with water
knowing that the specific gravity of mercury is about 13.6.
The question is how high would the water column be at atmosphere pressure?
So try to use some of the laws that we've discussed
about the way gas works and pressure works and measuring pressure
and see if you can figure out how high the water column would be
in a barometer made with water.
If you've written this out, hopefully what you did, looks something like this.
What we would do with the barometer,
first of all is again, let's just remember what a barometer is.
We have some liquid which we usually use mercury.
Then we have a tube inserted in this liquid.
The pressure in this case in the atmosphere
will push down and this will raise the liquid to be a certain height in this tube here.
So we have a particular height
and the fluid whatever it is whether mercury or water
will be exerting its own pressure downwards,
so all we have to do is equate.
We'll say that the pressure say in this case from the water
has to be equal to the pressure from the atmosphere.
We already know an expression for the pressure from the water,
this is rho gh, the height of the water,
where in this case we will be talking about the density of water.
This has to be equal again to the atmosphere pressure.
So this isn't so hard to solve, we can say that the height of the water would go to
is equal to the atmosphere pressure
divided by the density of water times the gravitational acceleration.
Similarly we knew, so this is the height of water
but we already know that the height of mercury which I'll call hm,
you might often see this written as Hg instead from mercury as we saw,
would be the exact same thing except we would use the density of mercury
times the gravitational acceleration.
So the reason that I wrote both of these
the height of the water expression and the height of the mercury expression
is that on this right-hand side are this height of mercury,
this is something that we know something about.
We already know the height to which mercury rises in a particular column.
So let's re-write this height of water since we wanna try to find the height of water
using the things that we already know.
So we know that the height of the water
which is equal to the atmosphere pressure
divided by the density of the water times g,
would be equal to and what I would like to do here
since we have the density of water written here.
But I would like not to actually use that expression
and we know the density of mercury is use what were given in the problem
which is that the density of mercury divided by the density of the water.
In other words, the specific gravity of mercury is equal to about 13.6.
So this were given in the problem, so we would like to write something in terms of this expression,
the ratio of the density is rather than just what we have here which is just the density of the water.
So let's do that. And the way we would do that in practice
is to simply multiply the numerator and denominator of this,
by the density of mercury and of course since we're doing this to the numerator and denominator
we haven't change our expression, we simply re-written it.
So now we can see that I've written this as the density of mercury
over the density of water times the atmosphere pressure
divided by the density of mercury times g.
So you see what I've done here, don't be confused at the Ws and Ms.
I moved the density of water out to the front
so that it's now here and I moved the density of mercury into the expression
so that it is now here because this expression we already know,
we already know what this is exactly, this is the height of a mercury column as we saw from this.
And we also know at this expression is because this is the specific gravity of mercury
as we saw from this and so now we're ready to plug in exactly the things that we know.
We know that this specific gravity is 13.6.
We know that the height of a mercury column is 760 millimeters.
And all we have do is plug in these numbers
and this if we multiply this out you should hopefully get something like 10.336 millimeters.
Where to understand the reason why we don't ever make water barometers,
this is about 10.3 meters as the height of the water would go to.
So it's clear from an example like this,
not only a little bit more about how a barometer works
but also why we needed to find such a heavy fluid,
in this case liquid mercury which is a metal and still liquid at room temperature?
Why we needed to find such an exotic
and heavy fluid before we can make practical tools like this
like a barometer or the height of the liquid would go to
isn't 10 meters or over 30 feet tall?