Before we move on, I just wanted to make sure that you
understood that last point that I made at the end of that
last video.
We said that the pressure inputting into this, that we
could view this cup with a hole in it as essentially a
pipe, where the opening on the top of the cup is the input to
the pipe, and this little mini-hole is the output to the
pipe, and we said that this is a vacuum.
Let's say this is vacuum all around.
I know when I drew it last time, I closed it, but we have
a vacuum everywhere.
Since there's a vacuum everywhere, the pressure at
this point P1 is equal to zero.
The point I wanted to make is because we have a hole here,
the pressure at that point at P2 is also equal to zero.
You can almost view it as maybe the atmospheric pressure
at that point, but since we're in a vacuum,
that pressure is zero.
That might have been a little confusing to you, because you
said, well, wait, I thought at depth, if I had a point at
that same height, that I would actually have a pressure at
that point of rho gh.
That's true.
That's completely true.
You do have an innate pressure in the liquid at that point of
rho gh, and actually, that's what's causing the
liquid to come out.
But that's actually taken care of in the potential energy
part of the equation.
Let me rewrite Bernoulli's equation.
The input pressure plus rho g h1 plus rho V1 squared over 2
is equal to the output pressure plus rho g h2 plus
PV2 squared over 2.
I think you understand that this term is pretty close to
zero if the rate at which the surface moves is very slow if
this surface area is much bigger than this hole.
It's like if you poked a hole in Hoover Dam, that whole lake
is going to move down very, very slowly, like 1 trillionth
of the speed at which the water's coming out at the
other end, so you could ignore this term.
We also defined that the hole was at zero, so the height of
h2 is zero.
It simplified down to the input pressure, the pressure
at the top of the pipe, or at the left side of the pipe,
plus rho gh1.
This isn't potential energy, but this was kind of the
potential energy term when we derived Bernoulli's equation,
and that equals the output pressure, or the pressure at
the output of the hole, at the right side of the hole, plus
the kinetic energy PV2.
It's the kinetic energy term, because it doesn't actually
doesn't add up completely to kinetic energy, because we
manipulated it.
I just wanted to really make the point that
is definitely zero.
I think that is clear to you, because we
have a vacuum up here.
The pressure at that point is zero, so we can ignore that.
The question is what is the pressure here?
This pressure is zero, because we have a vacuum here.
If I were to say that the pressure over here at this
hole is equal to pgh, then I would have the situation where
pgh is equal to pgh plus PV squared over 2.
What does that mean?
When I say that that pressure at the output of the pipe is
pgh, that means that I'm applying some
pressure into that hole.
Essentially, that pressure I'm applying into the hole is
exactly just enough offset to offset the
pressure at this depth.
Because of that, none of the water will move.
You could imagine that if this is the hole, let's say that's
the opening of the hole, and I have some water particles, or
some fluid particles, let's say that these are the atoms,
we're saying innately at any point that there is a pressure
at this point that's equal to rho gh, but this is P2.
How much pressure am I exerting on
this end of the hole?
If I exert rho gh at this end, then these molecules that were
just about to exit the hole aren't going to exit, because
they're going to get the same pressure from every direction.
What we said in the last video, and I really want to--
because this is a subtle point-- is that the outside
pressure, being on the outside part of the hole, is zero, and
because of that, we end up-- this term is zero, and we
essentially end up with that the change in the potential
energy all becomes kinetic energy, which is something
we're familiar with from just our kinematics
and our energy equations.
With that out of the way, let me do another problem.
Actually, I will do that next problem in the next video,
just so we have a clean cut between videos.
See you soon.