Water pressure depends only on depth, not the container shape.
Here we have three containers of the same depth.
The pressure at the bottom of each container is exactly the same for all three containers
because they have the same depth, even though there's much more water in this one, weighing
a lot more than this one.
The pressure, and therefore the force, on the bottom of these containers is exactly
the same.
How is that possible?
Let's take a look at the simplest example first a container with parallel sides like
this one with water inside and compare it to a stack of wooden blocks.
I'm better drawing cross-section views, so that's what I have here.
This block weighs two pounds, so it exerts a downward force of two pounds on the block
below it.
This block is not moving, so there's a counter-balancing force of the lower block pushing against the
upper block, also two pounds.
These two blocks together weigh four pounds, so they exert a force of four pounds on this
block, and likewise, this block exerts a counter-balancing force of four pounds.
Finally, the three blocks together weigh six pounds.
They exert a force of six pounds on the table, and the table exerts a counter-balancing force
of six pounds back.
Now let's look at the water.
This block of water weighs two pounds and exerts a force of two pounds downward on the
water below it, and the water below has a counter-balancing force upward.
The forces are balanced throughout the water.
The water also exerts a sideways force on the walls of the container outward, and the
wall exerts a counter-balancing force against the water.
Now, the forces against the wall are always perpendicular to the wall.
We're interested in the vertical force at this point, so we could ignore the sideways
force for now.
This much water weighs four pounds and exerts a force of four pounds downward on this block
of water, and again, there's a counter-balancing force.
Finally, this whole block of water exerts a total force of six pounds on the bottom
of this container, and the bottom of the container has a counter-balancing force, again, of six
pounds.
Now that's a total force on the bottom of the container.
What is pressure?
Pressure is force divided by area.
Let's say that the bottom of this container measures 4 by 4 inches or 16 square inches.
In that case, the pressure is 6 pounds divided by 16 square inches.
That's about 0.4 pounds per square inch, rounding to the nearest tenth.
What does that mean?
That means if you open a hole at the bottom of the container that measures 1 by 1 inch,
you need to apply a force of 0.4 pounds in order to prevent the water from coming out.
The water pushes in all directions, so if you open a hole on the side, it's the same
story.
You need to apply a force of 0.4 pounds to stop the water from coming out.
It even works upward if you have an opening that curves upward.
You need to apply 0.4 pounds of force to keep the water from gushing out.
Now let's look at the same thing using SI units, or metric units.
A 1 kg mass exerts a downward force of 9.8 Newtons.
I'll round that off to 10 Newtons.
Newtons is a measure force.
Kilograms is a measure of mass.
Mass is resistance to acceleration, whereas force, Newtons is the force.
So in Newtons, 10 N down for the first one.
There's a counter-balancing force of 10 N; 20 N down, counter-balancing force of 20 N;
and finally 30 N at the bottom.
Now let's take a look at the water.
This block of water weighs 10 N. The counter-balancing force is 10 N.
For this block of water 20 N, 20 N; and finally at the bottom of the container, 30 N of water
exerts a downward force of 30 N. The bottom of the container exerts a counter-balancing
force of 30 N.
Now again, pressure is force divided by area.
The total force is 30 N. Let's say the bottom of the container measures 10 by 10 cm or 0.1
meter by 0.1 meter, which is one hundredth of a square meter, so the pressure at the
bottom of this container is 30 N divided by 0.01 square meter, equals 3,000 Pascals.
One Pascal means one Newton per square meter, or equivalently, 3 kiloPascals; 3,000 Pascals
is the pressure at the bottom of this container.
Now for each 10 cm of depth, we have 1 kPa of pressure.
The pressure increases in a linear fashion as you go deeper and deeper inside container:
1 kPa at 10 cm deep, 2 kPa at 20 cm deep, and 3 kPa at 30 cm deep.
That's 3 kPa at the bottom of the container.
Ok now let's look at this equivalent container with the same area at the bottom.
I maintain that the pressure the bottom is still 3 kPa.
The extra water does not exert any more pressure on the bottom.
Why not?
If we're looking at a wooden block, this is certainly not true.
Say we have a block like this weighs a certain amount, and a bigger block with the same area
at the bottom, it will exert a greater force and therefore have a greater pressure, that
is, force per area at the bottom.
That's because this ... The only counter-balancing force is at the bottom.
The weight of this is transmitted through the block to the bottom.
Now that's for a block of wood.
Water is different.
It cannot transmit forces through itself other than pressure because it has no strength.
That is, you cannot twist or bend water; it just flows into and fills the container, and
it exerts a force perpendicular to the container, so the downward force of this additional block
... pushes against this container, not against the bottom, so we have the same pressure at
the bottom.
Now you may not be totally convinced, so let's go outside for a live demonstration that will
help shed some light on this.
Here I'm simulating a liquid by putting a bunch of rice.
Now the rice not exactly a perfect model.
It's an approximate model because the rice has friction.
Imagine if the rice grains didn't have friction.
What would happen?
You can put something in.
You can put a solid thing in there and it stays put because the forces on all sides
of the piece of cardboard are the same.
You put it in any angle, you know, tilt it or whatever, and it doesn't move because the
forces are balanced.
Now imagine.
We take ... I have a piece of cardboard here that will fit in snugly here.
I'll push that all the way down.
Again, the forces are balanced now.
What happens if we empty the rice from one side, so I'll do that right now.
Cover up one side, pour out the rice.
Now you see that the rice flattens out.
If there were no friction, it would not form a pile at all.
It would flatten to single layer.
Now what happens ... oops!
Well if you imagine that ... push this back and hold it here.
It exerts the same force against the cardboard as it would if there was rice in there.
I can pour the rice back and see that.
So it's the same force on either side whether or not this side has any rice in it.
Sorry about the airplane noise.
So we have this wide container full of water.
Let's divide this container into sections, like this, and insert a waterproof cardboard
wall in here, and here.
The forces are balanced on both sides of this wall, just like in the rice container, so
this piece of cardboard doesn't move.
It just sits there.
Now imagine we glue the cardboard at the bottom here, and here, and empty the water out up
from the sides.
Now you see this water inside this section of the container is now exerting pressure
against this wall that's exactly the same as it was against the water, but now this
wall is holding back that force instead of this water.
There's stress on this wall and this joint, and if it's not glued well, it's going to
collapse, just like it did in the rice container, but if we got a strong container here with
a strong joint, this water will exert the same exact same force as it did against this
water when there was no wall.
It's same for this side.
So we now we have the exact same situation we had before with parallel sides.
The depth ... the pressure depends only on depth.
And the water when you pour it into this container, it's not at equilibrium, but it settles down
until the pressure is constant at each depth: 1 kPa here, 2 kPa here, and 3 kPa at the bottom.
Now what about this container?
It has less than 30 N of water, and how can, say, 12 N of water exert a downward force
of 30 N on the bottom of this container?
To find out, let's look at the original container first with parallel sides, with 30 N of water.
Imagine we put two pieces of cardboard in here, like this.
Again, it doesn't move.
The pressure is balanced on all sides.
Now let's add two more pieces of cardboard down here, and we see that again, the forces
are balanced, although vertically this time.
We have the weight of this water pushing down on this piece of cardboard, and then the counter-balancing
force of this water under pressure pushing upward.
Again, the forces are balanced.
Now let's again say we glue these walls together and empty the water on the outside, and you
see that this piece of cardboard is now pressing down on the water with the same force that
it did when there was water inside.
Now there's again this joint.
There's a stress on these joints.
If you didn't glue it well, this is going to collapse and shoot upward.
But if it's strong, you've got the same force coming down and the water inside is pushing
up with the same force as it did when there was water in here.
This little piece of cardboard is taking the role of this big piece of water here.
So we have 12 N of water pressing down, and we must have 18 N of additional force pressing
down for a total of 30 N of downward force on this water.
That's why we can have 30 N of force on the bottom of this container, even with only 12
N of water.
The other additional force comes from the container pushing downward, and of course,
we have a counter-balancing force.
This explains why we have 30 N downward force inside the container, but outside the container,
resting on the table, we only have the weight of the water inside, 12 N, plus the weight
of the container itself.
The counter-balancing force is the water pressure upward against this container.
That's what, 18 N upward, to reduce the total weight of this container on the table.
So for any size or shape container, the depth ... the pressure depends only on depth: 1
kPa at this depth, 2 kPa at this depth, and 3 kPa on the bottoms, that are even at the
same depth.
If you're not totally convinced, you could punch a hole in the bottom of each container
and connect them with a pipe, and you would see that no water flows.
If you would initially fill the pipe with water and connect it up, no water flows.
If the pressure here was greater than the pressure here, water would flow from here
to here, but it doesn't.
As you know, the water seeks its own level, so no water flows.
The pressure at the bottom of three containers is exactly the same.
So I hope that helps you understand why the pressure depends only on depth and not the
shape of the container.