question I'm facing is and to figure out the work that we need to pump all this water from this tank
to the top of the tank so the water can be out in this video let me show you what I would do if
I just want to get this question right on the test okay first of all of course I don't know what it's
work and we know that work is defined to be force times distance and let me write down a distance
first okay so it's this x times force and what it's force force is just pretty much like the
weight right and now how do we calculate weight we have to care about two main things the first
thing is the density and the second thing is the volume right but for the density is kind of tricky
because were two kind of density the first one is what we call the weight density and the other
one's called mass density but it's kinda sad you know bizarre let me just write down density okay
so this tens times density and now if I just put this down if you just don't watch my whole
video if you guys don't listen to us you argue in the comment section that this is not correct but
you know what it's okay density times volume this right here will calculate the weight but we have
to be careful with the density part especially your the pay attention to what kind of units are
we in right here let me just indicate this for you guys in this case we are using meters that means
we're in the SI unit and whenever we are in the SI unit be sure that we need to multiply the 9.8 okay
well of course that's like for the gravity but in this case that would be the mass density but once
again let me just put down the density right here if we're using like say 2 feet 10 feet 4 feet if
we are in the u.s. unit we don't have to multiply by that 9.8 because that would be considered
equate density already right but remember we are pumping a water so you also have to know the magic
number which is 1,000 in the that unit for the mass density of the water anyway here we go let's
figure out the volume first okay that's usually the hardest part because you're to do geometry
you do some algebra and were the first to do the volume we have to come back here and I'm gonna set
up a reference frame and let me just put on the x-axis down below here so that everything will be
positive and I will put the y-axis right in the middle like this okay and the stylus will make a
horizontal cut and because this was a cone once we make a horizontal cut as you can see we will
have a cylinder right this right here it's a small disc and it's arguably a cylinder just like a CD
right it's pretty thin and we are to first record that how do we find out the volume of a cylinder
and now we'll figure out everything from right to left the thickness it's equal to the differential
okay utility whatsoever so we come back here as you can see here is the small cut that we made
right it's just a small disc and you will see that the thickness right here is just a small
change amount in the y-axis right so this will be T not es dy and that's the thickness and now let's
figure out the volume so let's put down the Y first next how to figure out the radius well based
on this picture as we can see the radius of this disc it will be from here to here which is going
to be the x-value because that's the horizontal distance okay so the radius is X so you have X
square and the next of course will be multiplied by the pie and now this right here represents the
volume of this disc we are not done yet but that's just continue okay next we are going to figure out
the density how can we figure out the density well it's just some constants that you to look up it
depends on what kind of things that you have okay right here we are talking about water so the mass
density of the water it's 1,000 and let's not put on any unit yet as nice you are making sure that
all the units are good you'll be ok as nice you don't have that say 10 meters and then all of a
sudden you get 2 centimeters and all of a sudden you get like four yards whatsoever that would be
a pretty messed up situation but usually that will happen okay anyway so the mass density of
the water that's water not gasoline will not milk whatsoever it's 1,000 right for water and
once again when wherever we're in the SI unit we have to remember to multiply by 9.8 because
of the gravity okay so we multiply by 9 point a so this right here will take care of the density part
of y but we have in the tank next we are going to figure out that this turns for the distance
it's a following we have to figure the distance from here all the way to the top of the tank so
the water can be out well based on this picture that we have distance right here or from here to
here is the Y value because that's the vertical distance the horizontal distance was the X okay
but I care about the distance from here to here so that water can be out ok well the whole thing
is 10 this much is why already which implies that from here to here the distance is just 10 - why
isn't it so let me write this down 10 minus y and we multiply by 10 - y right here this represents
the work that we need to pump out this much of the water from here to the top of the tank okay
and how many of these kind of things do we have a lot here is the first one all the way up to
here this is the time we are to integrate to add them up okay so we integrate over well the Y power
F here is zero up here originally was 10 for the whole thing but the first two meters on the top is
empty so from here to here is only 8 therefore we have to integrate from 0 to 8 and this will pretty
much calculate the work for you and fortunately notice that X it's not invited in the wide world
so we had to do more math okay and here is where the X and y-axis comes in because we can do some
know geometry by using algebra okay if you look at this picture unless died here this is just a
straight line and of course we know how to find an equation of a line this point here based on the
choice of our reference frame is through zero and then this point as we know from here to here the x
value would be four and then the Y value is from here to here which is 10 so that's 40 comma 10
of course we have to get a slope first so that's an equal y2 minus y1 which is 10 minus zero over
x2 minus x1 which is 4 minus 0 10 over 4 pretty much which is 5 over 2 and then next we can use
the generic question Y minus y1 equal to M times X minus x1 and I know I could have used Y is equal
to MX plus B but let me tell this right here will cover everything especially if you don't get like
a nice chrome maybe just a half of the connector all right so I like to show you guys this for
consistency purpose anyway let me label this one as x1 y1 so now you see Y is just Y minus y1 which
is 0 equal M is Phi 1/2 times X is just X minus x1 which is 0 and now this is just y equal this comes
down only which is Phi over 2 X right but I want to isolate the X so in other word X would be after
we multiply both sides by 2 over 5 we can X equal 2 over 5 X and now I will just have to plugging
this into this X and I will be done for the setup and you guess what do the work of not maybe use a
calculator will do it by hat anyway work is equal to from 0 to 8 10 minus y times 9.8 times 1000
and then multiplied by we have the PI is still pi and then for the X we are going to open the
parentheses and put the same which is 2 over 5 Y and then square that at the end we of course have
this D Y this integral will calculate the work for you but you also have to do the work on your own
to figure out the value for this leave a comment down below who ever commanded correctly first I