Pressure is a scalar; the units of pressure in the SI

system are pascals:

1 Pa = 1 N/m2.

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.