in this problem we want to calculate the
position of the force that is exerted
due to water pressure on a straight wall
in the previous problem we had
calculated the magnitude of this force
and so we have a wall that looks like
this more or less we had water like so
on this side of the on this side of the
wall not quite like so yes and we had
calculated the magnitude of this force
so we had F here F P which was about 6
meganewtons and what we want to do now
is we want to calculate how far away
from the bottom this force applies
we're gonna call this distance R F like
so and the way we're gonna do this is we're
in calculate the moment that is exerted
around this bottom hinge we’re gonna say the
bottom of the wall is like a hinge or
rotating hinge and we're gonna calculate
the amount of torsion that is exerted by
this force by this by the water pressure
on the wall and so we'll be able to know
we will calculate the moment due to
pressure that is exerted by water
on this wall with the force and the
moment we can calculate the radius
because we know that the moment divided
by the force is equal to the radius so
it's mostly an engineering problem and
this engineering problem is solved very
much like the force problem was solved
so let's take a look at the side of the
wall where we have here the wall seen
from the side and we still have the
water surface which is over here inside
this water we said we would
position ourselves to determine pressure
where the coordinate z, that would start at
the water surface and we would be at
this height we would be here at this
position away from the bottom here and
the coordinate R along the wall this R
goes from 0 here up to R max at the top
and we would be looking here at a little
bit of wall at a little strip of wall
horizontal strip of wall with thickness
DR like so
well what is the moment that is exerted
on this tiny bit of surface we have a
force and this force is aligned
here it is called DF and this force is
the same thing as a tiny moment here DM
exerted about the bottom hinge or the
bottom edge of the wall so how do we
calculate the moment well we're going
to calculate the moment the same way we
calculated the force we’re gonna say the total
moment due to pressure is
the sum of a great deal of little
moments actually an infinite number of
little moments it's gonna go like this
DM DM due to pressure let's just call
the moments M it's been probably easier
to to write so the moment is the
integral of a whole sum of tiny moments
and every time each of those tiny
moments is the radius away from the
bottom at which the tiny force applies
multiplied by this time force DF and
this force DF is relatively simple to
express we've done this before when we
calculated the force so let me carry out
this integral below so I have a bit more
space and we're gonna write it like this
we’re gonna say the moment is the integral of
r DF and it is the integral of r DF
being now pressure multiplied by the
tiny bit of area d s.Pressure is going
to be guided by the distance away from
the top surface at which we are it's
going to be Rho G Z so let's rewrite this
like this we can say this is the
integral of R Rho G  Z here and
then D s tiny bit the surface is a tiny
bit of height that we have here
multiplied by the width of the whole
panel and the width would be here this
21 meters that we have here on the
bottom bottom right of the screen as I'm
going to have L 1 which is 21 meters and
then D R over here from where to where
do we carry out this integral we carry it out from R is equal to 0 at the bottom
right all the way up to R is equal to
Rmax and so let's plot the maximum
values that we have here this is going
to be here R max like this this
is a distance this is not a coordinate
so we carry out the integral from R is
equal to 0 to R is equal to R max like
this ok now remains the same problem
that we had before which is that we have
an integral here which is determined in
terms of dr but inside this we have a
Z and so we're stuck with the Z here
until we transform the Z or we
express the Z in terms of R how do
we do this we have to look at our
diagram over here and we have to express
this in in this diagram as a function of
R and the way we deal with this is we
say Z is equal to Z max which is the
maximum value of Z which happens to be
the top value here like so this would be
Zmax and then from Z max, Z can only
decrease it starts as nZmax and then as
I grow up with our here it is going to
decrease every time by an amount which
is equal to R and so I have Z is equal
to Z max minus R the coordinate R like
this so this is a geometrical constraint
that translates the constraints of
gravity positive downwards with the
constraints of following my panel which may
follow another direction and this
equation here this bit here we're going
to include inside this integral here so
let's carry this out M is equal to now
the integral from 0 to R max of our Rho
G, Z now becomes Z max minus R like this and then we have L 1
and we have dR
let's move the terms that can be moved
outside the range
and those are all that do not depend on
R and there was our rho G and L one so
I'm gonna write them out and then we
carry out the integral from zero to R
max of R Zmax yes and then minus R
squared like this and this is done with
respect to DR like so okay this is
done now all the difficult parts are
there are done and we just left with
some basic algebra this is not a very
difficult integral so let's see if I can
carry it out without goofing up we have
Z max R it's going to integrate
as Z max one-half
of R squared and then minus R squared is
going to integrate as minus one-third of
R cubed minus this primitive function we
evaluate between R max and zero and so
we can write the final result now as
being the moment is equal to…
…
moment we put numbers in this and then
let's see how we can calculate the
radius based on this first bit numbers
first and then we have density he has 10
to the power 3 1000 clear as per meter
cube
gravity is 9.81 me write this is not
very clean
harder to write when I'm at the bottom
of the screen like this I multiply by l1
l1 happens to be here at the bottom
right of the screen
…
meters yes and we rearrange this to type
it in more easily so I'm going to say
this is 9.81 times 10 to the power 3
times 21 times 8 cube multiplied by 1/2
minus 1/3 this is not necessary but it
makes typing into the calculator easier
and reduces the chances that I mess it
up so I did this for you before and
this turns out to be 1 point 7 5 8 times
10 to the power 7 Newton meters this is
the unit of the moment yes all right
it's very hard to estimate how big or
small a number this is
pressure forces
tend to be very large but we can check
the or we can have an idea of how bad or
good the result is when we calculate the
final radius so let's calculate the
radius now let me go back here to the
top and take a look again at the radius
that we want to calculate we want to
calculate from the previous page yeah
the distance away from the bottom hinge
at which the force applies and for this
the equation that we want to write is
this RF is equal to the moment divided
by the force so we're going to take the
value of the moment divided by the value
of the force and we're going to get the
value of the radius
this is gonna be
here the radius is going to be the
moment divided by the force the moment
…
this is in Newton meters and I divide
this by a force which is in Newtons
which happens to be six point five nine
two times ten to the power six Newtons
we calculated this previously and if
you take this and you get a value of two
point six six seven meters this is the
RF value here so we can briefly check
that this makes sense because we have an
8 meter height in total so I want
definitely this radius here to be less
than 8 meters I don't want it to be a few
millimeters and it turns out the force
here that we calculated before this
force exerts to point seven meters away
from the bottom. So this is how you
calculate the position of a force due to
pressure on a flat wall in fluid mechanics
