Physics 151: Lecture 29, Pg 1 Physics 151: Lecture 29 Today’s Agenda l Today’s topics çFluids...

Physics 151: Lecture 29, Pg 1

Physics 151: Lecture 29 Physics 151: Lecture 29 Today’s AgendaToday’s Agenda

Today’s topicsFluids under static conditions, Ch. 14.1 through 14.4PressurePascal’s Principle (hydraulic lifts etc.)Archimedes’ Principle (floatation)


FluidsFluids

At ordinary temperature, matter exists in one of three states

Solid - has a shape and forms a surfaceLiquid - has no shape but forms a surfaceGas - has no shape and forms no surface

What do we mean by “fluids”?Fluids are “substances that flow”…. “substances that

take the shape of the container”Atoms and molecules are free to move.No long range correlation between positions.

See text: 14.1


FluidsFluids What parameters do we use to describe fluids?

Density

V m

See text: 14.1

units :

kg/m3 = 10-3 g/cm3

(water) = 1.000 x103 kg/m3 = 1.000 g/cm3

(ice) = 0.917 x103 kg/m3 = 0.917 g/cm3

(air) = 1.29 kg/m3 = 1.29 x10-3 g/cm3

(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3


FluidsFluids

nF ˆpA A

n

Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface.

Force (a vector) in a fluid can be expressed in terms of pressure (a scalar) as:

AF

p

units : 1 N/m2 = 1 Pa (Pascal)1 bar = 105 Pa1 mbar = 102 Pa1 torr = 133.3 Pa

1atm = 1.013 x105 Pa = 1013 mbar = 760 Torr

= 14.7 lb/m2 (=PSI)

What parameters do we use to describe fluids?Pressure


When the pressure is much less than the bulk modulus of the fluid, we treat the density as constant independent of pressure:

incompressible fluid For an incompressible fluid, the

density is the same everywhere, but the pressure is NOT!

Pressure vs. DepthIncompressible Fluids (liquids)

Consider an imaginary fluid volume (a cube, face area A)The sum of all the forces on this volume must be ZERO as

it is in equilibrium: F2 - F1 - mg = 0

y1 y2

A

F1

F2

mg

p1

p2

0p

ApApFF 1212 Ag)yy(mg 12

)yy(gpp 1212

See text: 14.2


If the pressures were different, fluid would flow in the tube!

However, if fluid did flow, then the system was NOT in equilibrium since no equilibrium system will spontaneously leave equilibrium.

Pressure vs. Depth

For a fluid in an open container pressure same at a given depth independent of the container

p(y)

y

Fluid level is the same everywhere in a connected container, assuming no surface forces

Why is this so? Why does the pressure below the surface depend only on depth if it is in equilibrium?

Imagine a tube that would connect two regions at the same depth.

See text: 14.2


Lecture 29, Lecture 29, ACT 1ACT 1PressurePressure

What happens with two fluids?? Consider a U tube containing liquids of density 1 and 2 as shown:Compare the densities of the liquids:

A) 1 < 2 B) 1 = 2 C) 1 > 2

1

2

dI


ExampleExample

A U-tube of uniform cross-sectional area, open to the atmosphere, is partially filled with mercury. Water is then poured into both arms. If the equilibrium configuration of the tube is as shown in Figure on the right, with h2 = 1.00 cm.

Determine the value of h1.


ExampleExample

Figure on the right shows Superman attempting to drink water through a very long straw. With his great strength he achieves maximum possible suction. The walls of the tubular straw do not collapse.

(a) Find the maximum

height through which he can lift the water.


Pascal’s PrinciplePascal’s Principle

So far we have discovered (using Newton’s Laws):Pressure depends on depth: p = gy

Pascal’s Principle addresses how a change in pressure is transmitted through a fluid.

Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel.

Pascal’s Principle explains the working of hydraulic liftsi.e. the application of a small force at one place can result

in the creation of a large force in another.Does this “hydraulic lever” violate conservation of energy?

» Certainly hope not.. Let’s calculate.

See text: 14.2



Consider the system shown:A downward force F1 is applied

to the piston of area A1.This force is transmitted through

the liquid to create an upward force F2.

Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid.

F F

1

2

d

2d

1

A A21

2

2

1

1

AF

AF

1

212 A

AFF

F2 > F1 : Have we violated conservation of energy??

See text: 14.2



Consider F1 moving through a distance d1.

How large is the volume of the liquid displaced?

F F

1

2

d

2d

1

A A21

111 dV A

12 VV 2

112 A

Add

Therefore the work done by F1 equals the work done by F2

We have NOT obtained “something for nothing”.

12

11

1

21222 W

AA

dAA

FdFW

See text: 14.2

This volume determines the displacement of the large piston.


A1 A10

MdA

A2 A10

MdB

A) dA=(1/2)dB B) dA = dB C) dA = 2dB

Lecture 29, Lecture 29, ACT 2aACT 2aHydraulicsHydraulics

Consider the systems shown to the right.In each case, a block of mass M

is placed on the piston of the large cylinder, resulting in a difference dI in the liquid levels.

If A2 = 2A1, compare dA and dB.


A1 A20

MdC

A1 A10

MdA

A) dA = (1/2)dC B) dA = dC C) dA = 2dC

Lecture 29, Lecture 29, ACT 2bACT 2bHydraulicsHydraulics

Consider the systems shown to the right.In each case, a block of mass M is

placed on the piston of the large cylinder, resulting in a difference dI in the liquid levels.

If A10 = 2A20, compare dA and dC.


Archimedes’ PrincipleArchimedes’ Principle

Suppose we weigh an object in air (1) and in water (2).How do these weights compare?

W2?W1

W1 < W2 W1 = W2 W1 > W2

Why?

» Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object.

See text: 14.4


The buoyant force is equal to the difference in the pressures times the area.

W2?W1

)Ay-g(y)( 12 AppF 12B

liquidliquidliquidliquidB WgMgVF

Archimedes:

The buoyant force is equal to the weight of the liquid displaced.

The buoyant force determines whether an object will sink or float. How does this work?

y1

y2

Ap

1

p2

F1

F2

Archimedes’ PrincipleArchimedes’ Principle

See text: 14.4


Sink or Float?Sink or Float?

The buoyant force is equal to the weight of the liquid that is displaced.

If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink.

F mgB

y

We can calculate how much of a floating object will be submerged in the liquid:

Object is in equilibrium mgFB

objectobjectliquidliquid VgVg

liquid

object

object

liquid

V

V

See text: 14.4

Animation


The Tip of the IcebergThe Tip of the Iceberg

What fraction of an iceberg is submerged?

F mgB

y

liquid

object

object

liquid

V

V

%90kg/m 1024

kg/m 917V

V3

3

water

ice

ice

water

See text: 14.4


Lecture 29, Lecture 29, ACT 3ACT 3BuoyancyBuoyancy

A lead weight is fastened to a large styrofoam block and the combination floats on water with the water level with the top of the styrofoam block as shown.

If you turn the styrofoam+Pb upside down, what happens?

styrofoam

Pb

A) It sinks C)B) styrofoam

Pb

styrofoam

PbD)

styrofoam

Pb


ACT 3-AACT 3-AMore Fun With BuoyancyMore Fun With Buoyancy

Two cups are filled to the same level with water. One of the two cups has plastic balls floating in it. Which cup weighs more?

Cup I Cup II

(A) Cup I (B) Cup II (C) the same (D) can’t tell

See text: 14.4


ACT 3-BACT 3-BEven More Fun With BuoyancyEven More Fun With Buoyancy

A plastic ball floats in a cup of water with half of its volume submerged. Next some oil (oil < ball < water) is slowly added to the container until it just covers the ball.

Relative to the water level, the ball will:

water

oil

(A) move up (B) move down (C) stay in same place

See text: 14.4


Recap of today’s lectureRecap of today’s lecture

Chapter 14.1-4 PressurePascal’s PrincipleArchimedes Principle

Physics 151: Lecture 29, Pg 1 Physics 151: Lecture 29 Today’s Agenda l Today’s topics çFluids...

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