The pressure at a depth h below the surface of the liquid is
due to the weight of the liquid above it. We can quickly
calculate:
This relation is valid for
any liquid whose density
does not change with
depth.
2 1
2
Consider the tank shown in the figure. It contains a fluid
of density at rest. We will determine the pressure
difference between point 2 and point 1 whose
y-coordinates are a
p p
y
ρ
−
Fluids at rest
1nd , respectively. Consider
a part of the fluid in the form of a cylinder indicated by
the dashed lines in the figure. This is our "system" and
its is at equilibrium. The equilibrium condition is:
y
2 1 2 10 Here and are the forces ynetF F F mg F F= − − =2 1 2 1
1 1 2 2 1
exerted by the rest of the fluid on the bottom and top faces
of the cylinder, respectively. Each face has an area .
, ,
ynet
A
F p A F p A m V A yρ ρ= = = = −( )
( ) ( ) ( )
2
2 1 1 2 2 1 1 2
1 2 1 2
If we substitute into the equilibrium conditon we get:
0
If we take 0 and then and
The equation above takes the form:
o
o
y
p A p A gA y y p p g y y
y h y p p p p
p p gh
ρ ρ
ρ
− − − = → − = −
= = − = =
= +
op p ghρ= +
( ) ( )2 1 1 2p p g y yρ− = −
The difference is known as " "op p−Note : gauge pressure
A change in the pressure applied to an enclosed incompressible liquid
is transmitted undiminished to every portion of the fluid and to the
walls of the container
If an external pressure is applied to a confined fluid, the
pressure at every point within the fluid increases by that
amount.
This principle is used, for example, in hydraulic lifts and
hydraulic brakes.
This is an object submerged in a fluid. There is a net force on
the object because the pressures at the top and bottom of it
are different.
The buoyant force is found to be
the upward force on the same
volume of water:
Archimedes’ principle:
The buoyant force on an object
immersed in a fluid is equal to the
weight of the fluid displaced by that
object.
Consider the three figures to the left. They show
three objects that have the same volume ( ) and shape
but are made of different materials. The first is
made of water, the second
V
Archimedes' principle
of stone, and the third
of wood. The buoyant force in all cases is the
same: This result is summarized in
what is known as "
When a body is fully or partially subme
"
rg
b
b f
F
F gVρ=
Arhimedes' Principle
ed in a fluid�
When a body is fully or partially submerged in a fluid
a byoyant force is exerted on the body by the
surrounding fluid. This force is directed upwards
and its magnitude is equal to the weight g of the
fluid that has been displaced by t
b
f
F
m
�
We note that the submerged body is fig.a is at equilibrium
with . In fig.b and the stone accelerates
downwards. In fig.c and the wood accelerates
u
he body.
pw
ards.
g b g b
b g
F F F F
F F
= >
>
If an object’s density is less than that of water, there will be an
upward net force on it, and it will rise until it is partially out of
the water.
(a) The fully submerged log accelerates upward because FB > mg. It comes to equilibrium (b) when ΣF = 0, so FB = mg = (1200kg)g. Thus
1200 kg, or 1.2 m3, of water is displaced.
For a floating object, the fraction that is submerged is given
by the ratio of the object’s density to that of the fluid.
An object floating in equilibrium: FB = mg.
If the density doesn’t change—typical for liquids—thissimplifies to A1v1 = A2v2. Where the pipe is wider, the flow is
slower.
Conservation of energy gives Bernoulli’s equation:
Bernoulli’s principle:
Where the velocity of a fluid is high, the pressure is low, and where
the velocity is low, the pressure is high.
Using Bernoulli’s principle, we find that the speed of fluid coming
from a spigot on an open tank is:
ViscosityReal fluids have some internal friction, called viscosity.
Viscosity is an internal frictional force within fluids.