what is the pressure at the bottom of
the lake this is one of my favorite
exercises to do in class because
typically what I do in class is I
tell students "static fluid
pressure only depends on altitude" and
everybody looks at me like yeah of
course we know this we've seen it before
and then you give up this exercise in
class and see so many of the students
struggle with the cosine of the angle
right there
cosine or sine of 30 degrees and how do
I calculate the length there and so and
so forth well the answer is of course
four bars it's very easy to calculate so
let me show you how to how to solve this
problem you have the water surface on
top yes and then you have your point
which is down there and this point is at
a distance away from the from the top
which is measured with coordinate z at
the same time you have G gravity which
is pushing downwards what we have in
static fluid is that grad P is Rho G so
the gradient of
pressure grad P is Rho G
like this the change in space of
pressure is equal to density times
gravity and if G here is aligned with Z
then you can just take this 3d equation
and have it in only one direction and
this becomes here DP over DZ
is Rho G like so DP DZ is Rho G then
you can integrate and say then that it's
Rho G Z plus whatever value you
start at at z is equal to zero so we
call this P Zero here so if you have
here at Z starting on the water surface
at the water surface you have P
atmosphere this is P atm here and at
this point here you will have this patm
p0 here plus density times gravity times
whatever the distance away from the
top is so if you apply at a point
which
is shown just below here 30 meters down
inside water then you take data for
water 10 to the power 3 kilograms per
meter cubed multiplied by G which is almost
10, 9.81 times the z
value which is in this case 30 and
then you add pressure atmospheric which
is 1 bar which turns out to be 10 to the
power 5 Pascals and this I can move to
have a little space if you calculate
this gets you precisely 3.94 times
10 to the power of 5 Pascals yes and
we convert this into bar usually as
engineers and this is of course I'm
sorry I wrote this poorly this should
have been a 3 here this is a 3 like so
3.94 bar pressure at the bottom of the
of the lake and the rule of thumb is
that pressure inside static water
increases by about 0.1 bar per meter so
10 meters is 1 bar 30 meters is 3
bars more than the starting pressure so
here you go this is a very simple
problem and this is how you calculate
pressure at the bottom of any recipe let
me take the opportunity to add a few
more remarks with what this problem
actually means and what it what it's
meant to trigger a thought let's say you
have the water surface over here let me
keep it wavy like this and you're at
this point here just to be clear the
side walls that you may have on the side
here do not affect the pressure at this
point in any way as long as the water is
static yes and so this is true for this
wall but it's also true for this wall
like so you can make a wall like this
yes pressure here will still be Rho GZ
and you may even take the Atlantic Ocean
on the side and have huge lengths
towards the right it does not affect the
height you may even take here the space
to put a boat maybe a thirty thousand
ton boat on top of this as long as the
flow is zero so the water is static the
pressure at this point will not change
it remains the same just depends on the
attitude ah so here you go so this is
how you calculate the pressure depending
on depth inside a static fluid
