Lesson 9-1Fluids and Buoyant

Force

Define a Fluid

Matter is normally classified in one of three groups Solid, Liquid, Gas

Liquids share a commonality with gases in that they can flow and alter their shape as they flow

Solids do not share this property, they cannot flow nor can they readily change their shape; solids are not fluids

Definite Volume

Even though both liquids and gases are fluids, they have a distinct difference Liquids have a definite volume Say you have a one gallon gasoline can full of

fuel If you pour the entire can into a lawn mower fuel

tank, there will still be one gallon of gasoline

Indefinite Volume

Gases have neither definite volume or shape When a gas is poured, not only does it

change its shape to fit the new container, the gas expands to fill the new container

Density and Buoyant Force

Have you ever felt uncomfortable in a crowded room? Probably because there were too many people for the amount of space; the density of people was too high

Density is how much there is of a quantity in a given amount of space The quantity can be anything from people to cars

to mass or energy

Density and Buoyant Force

Mass density is mass per unit volume of a substance

When we talk about a fluid’s density, we are really talking about mass density

Mass density is represented by the Greek letter rho (ρ)

capital V for volume

The SI unit for mass density is kg/m3

ρ = mV

Pressure

Solids and liquids tend to be incompressible Their density change very little with changes

in pressure Gases are not completely incompressible

meaning their volumes are greatly affected by pressure

That is why there is no standard density for gas as there is for solids and liquids

Buoyant Force

Have you noticed that heavy objects seem lighter under water It is much easier to pick up a person in a

swimming pool than on dry ground That is because the water exerts an upward force

on objects that are partially or completely submerged

The upward force is called the buoyant force

Buoyant Force

Have you ever laid on a raft in a pool You and the raft experience a buoyant force,

which keeps both you and the raft afloat Because the buoyant force acts in the opposite

direction as the force of gravity, objects submerged in a fluid such as water have a net force on them that is smaller than their weight

Buoyant Force

This means they appear to weigh less in water

The weight of an object immersed in a fluid is the object’s apparent weight

Archimedes’ Principle

Imagine you fill a bucket to the top and drop in a brick What happens? Why? The total volume of water that overflows is the

displaced volume of water. The volume of the water is equal to the volume of the portion of the brick that is underwater

Archimedes’ Principle

The magnitude of the buoyant force acting on the brick is known as Archimedes’ Principle Any object completely or partially submerged in a

fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object

Archimedes’ principle can be written as

F m gB f=

Archimedes’ principle

Whether an object will float or sink depends of the net force acting on it

This net force is the object’s apparent weight and can be calculated as follows

F F F objectnet B g= − ( )

Archimedes’ principle

Now apply Archimedes Principle, using mo to represent the mass of the object

Rewrite using that idea

F m g m gnet f o= −

F V V gnet f f o o= −( )ρ ρ

Archimedes’ principle

A floating object cannot be denser than the liquid in which it floats

The expression for the net force on an object floating on the surface gives the following result

Or:

F V V gnet f f o o= = −0 ( )ρ ρ

ρρf

o

o

f

V

V=

Archimedes’ principle

We know the displaced volume of fluid can never be greater than the volume of the object

That means for an object to float the object’s density must be less than the displaced fluid’s density

If the volume of the object is equal to the volume of displaced liquid, the entire object is submerged

Floating Objects

For floating objects, the buoyant force equals the object’s weight Imagine a raft with cargo floating on a river

Two forces act on the raft and the cargo The downward force of gravity The upward buoyant force

Because the raft is floating, the raft is in equilibrium and the two forces are balanced

Floating Objects

This means for floating objects the force of gravity is equal to the buoyant force

As in

Notice for a floating object, the buoyant force can be found by using the first condition of equilibrium, Archimedes’ principle would be overkill in such a situation

F m gB o=

Apparent Weight

The apparent weight of a submerged object depends of density Think of or raft from earlier

Imagine a hole is punched in the raft The raft and cargo eventually sink below the water’s

surface The net force on the raft and cargo is the difference

between the buoyant force and the weight of the raft and cargo

Apparent Weight

As the volume of the raft decreases, the volume of displaced water also decreases, as does the magnitude of the buoyant force

This is shown with

After the raft becomes completely submerged, the two volumes are equal

Notice that both the direction and the magnitude of the net force depend on the difference between the density of the object and the density of the fluid in which it is immersed

F V V gnet f f o o= −( )ρ ρ

F Vgnet f o= −( )ρ ρ

Apparent Weight

If the object’s density is greater than the fluid density, the net force is negative (downward) and the object sinks

If the object’s density is less than the fluid density, the net force is positive and the object rises to the surface and floats

If the densities are the same, the object floats, but underwater

A simple relationship between the weight of a submerged object and the buoyant force on the object can be found by considering their ratios as follows

m g

Fo

B

o

f

= ρρ

Lesson 9-2Fluid Pressure and

Temperature

Pressure

You experience pressure everyday When you dive to the bottom of a swimming pool When you drive up a large hill When you ride in a airplane

Pressure

Pressure is force per unit area The fluids above are exerting force against your

eardrums, so your ears want to ‘pop’ to adjust for the pressure change

Pressure is measure of how much force is applied of a given area. It can be written as

PF

A=

Pressure

The SI unit of pressure is the N/m2 which is called a Pascal (Pa) A Pascal is a very small amount of pressure. Air pressure at sea level is about 105 Pa which is

1 atm The total air pressure in a typical automobile tire

is about 300000 Pa or 3 atm.

Fluids Exert Pressure

When you use a bicycle pump to put air into a tire, you apply force on the piston, which exerts a force on the gas inside the tire. The gas pushes back on the piston and on the

walls of the tire. The pressure is the same throughout the volume

of the gas

Hydraulic Pressure

An important use of fluid pressure is the hydraulic press Garages can use a motor to generate a large

force over a small area to provide a force to equal a large area, such as lifting a car

F

A

F

A1

1

2

2

=

Pressure and Depth

Pressure varies with depth in a fluid As a submarine dives into the water the pressure

of the water against the hull of the sub increases. Water pressure increases with depth because the

water at a given depth must support the weight of the water.

The weight of the entire column of water above an object exerts force on the object.

Pressure and Depth

The column of water exerting force has a volume equal to Ah were A is the cross sectional area and h is the depth.

The pressure at a depth caused by the weight of a volume of water can be calculated as

PF

A

mg

A

Vg

A

Ahg

Ahg= = = = =ρ ρ ρ

Gauge Pressure

This pressure is referred to as gauge pressure. It is NOT the total pressure at this depth because atmospheric pressure is applying pressure at the surface.

Absolute pressure P is calculated

Po is atmospheric pressureP P hgo= + ρ

Atmospheric Pressure

Atmospheric pressure is pressure from above The weight of the air pushing down on the earth

and the bodies on earth is known as atmospheric pressure.

Atmospheric pressure is actually quite large, assuming a SA of 2 m2, ATM is 200,000 N.

How can our bodies withstand such an incredible force without being crushed? Bodies are in equilibrium, fluids inside push back

with the same force creating a state of balance

Lesson 9-3Fluids in Motion

Ideal Fluid

Ideal fluid model simplifies fluid flow analysis Many fluid features are considered by studying an

ideal fluid No real world fluid is an ideal fluid, but an ideal

fluid does showcase many properties of real world fluids

Idea Fluid

We assume an ideal fluid is incompressible, meaning the density always remains constant

We also assume an ideal fluid is nonviscous Viscosity refers to the amount of internal friction in

a fluid A fluid with a high viscosity tries to bond with its

container and flows more slowly than a low viscous fluid

Ideal Fluid

Another property of an ideal fluid is ideal flow We assume the velocity, density and pressure at

every point in the fluid is constant. This is known as non-turbulent flow, there can be

no undertows or rip currents present in the moving fluid

Conservation Laws of Fluids

If a fluid flows into a pipe, the mass that flows into the pipe must equal the mass exiting the pipe, even if the diameter of the pipe changes

x2

x1

V2

V1

A2

A1

Conservation Laws in Fluids

This can be shown as Recall and and recall

Both the time interval and density remain constant though (ideal fluid), so they are cancelled and we are left with

m m1 2=m V= ρ V A x= ∆

ρ ρ1 1 1 2 2 2A x A x∆ ∆= vd

t=

ρ ρ1 1 1 1 2 2 2 2A v t A v t=

A v A v1 1 2 2=

Conservation Laws in Fluids

This is referred to as the continuity equation where A shows two different cross sectional areas and v shows two different velocities

A v A v1 1 2 2=

Conservation Laws in Fluids

As the cross sectional area increases, the velocity slows, and as the cross sectional area decreases, the velocity increases to flow the same volume of water

The flow rate remains constant regardless the diameter of the pipe

Conservation Laws in Fluids

The expressions for conservation of energy in fluids differs slightly from our previous form studied in chapter 5

The reason is that fluids also exert pressure, so the conservation equation must take into account the pressure of the fluids

A change in pressure can be related to the transfer of energy into or out of the volume. We must account for this energy.

Conservation Laws in Fluids

As a fluid moves through a pipe of varying cross sectional area and elevation, the pressure and speed can change, but the total energy does not.

Bernoulli’s equation explains such a situation constant, that is to say conserved

P v gh+ + =1

22ρ ρ

Conservation Laws in Fluids

We can set up a version of the equation to compare the energy of a fluid at two different points in a pipe

P v gh P v gh1 12

1 2 22

2

1

2

1

2+ + = + +ρ ρ ρ ρ

Special Case 1

Lets say the fluid is not moving and the initial height is zero

a pressure function of depth

P P gh1 2 2= + ρ

Special Case 2

Lets say a fluid is flowing through a horizontal pipe with restriction

Since

This expression implies that if at some point in the flow, then

P v P v1 12

2 221

2

1

2+ = +ρ ρ

h h1 2=

v v2 1>P P2 1<

Bernoulli’s Principle

Swiftly moving fluids exert less pressure than slowly moving ones.

Reason why a curve ball curves, a plane can fly, a soccer player can ‘bend it like Beckham’ and why homes are ripped to shreds during hurricanes and tornadoes