Lecture Lecture VIIIVIII

General Physics (PHY 2130)

• Solids and fluids density and pressure buoyant force Archimedes’ principle Fluids in motion

http://www.physics.wayne.edu/~apetrov/PHY2130/

Lightning ReviewLightning Review

Last lecture:

1. Rotational dynamics torque and angular momentum two equillibrium conditions

Review Problem: A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational inertia and her angular speed increases so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be

1. the same.2. larger because she’s rotating faster.3. smaller because her rotational inertia is smaller.

Example:Example:

Given:

Moments of inertia:

I1 and I2

Find:

K2 =?

LIKErot 2

1

2

1 2

Rotational kinetic energy is

We know that (a) angular momentum L is conserved and (b) angular velocity increases

Thus, rotational kinetic energy must increase!

Solids and FluidsSolids and Fluids

QuestionQuestion

What is a fluid?

1. A liquid2. A gas3. Anything that flows4. Anything that can be made to change shape.

States of matter: Phase Transitions

ICEICE WATERWATER STEAMSTEAM

Add heatAdd heat

Add heatAdd heat

These are three states of matter (plasma is another one)

These are three states of matter (plasma is another one)

States of MatterStates of Matter

►SolidSolid►LiquidLiquid►GasGas►PlasmaPlasma

States of MatterStates of Matter

►SolidSolid

►LiquidLiquid►GasGas►PlasmaPlasma

Has definite volumeHas definite volume

Has definite shapeHas definite shape

Molecules are held in specific Molecules are held in specific location by electrical forces and location by electrical forces and vibrate about equilibrium vibrate about equilibrium positionspositions

Can be modeled as springs Can be modeled as springs connecting moleculesconnecting molecules

States of MatterStates of Matter

►SolidSolid

►LiquidLiquid►GasGas►PlasmaPlasma

Crystalline solidCrystalline solid Atoms have an Atoms have an ordered structureordered structure Example is salt (red spheres are Example is salt (red spheres are

NaNa++ ions, blue spheres represent ions, blue spheres represent ClCl-- ions) ions)

Amorphous SolidAmorphous Solid Atoms are arranged Atoms are arranged randomlyrandomly Examples include glassExamples include glass

States of MatterStates of Matter

►SolidSolid►LiquidLiquid

►GasGas►PlasmaPlasma

Has a definite volumeHas a definite volume

NoNo definite shape definite shape

Exist at a higher temperature than Exist at a higher temperature than solidssolids

The molecules “wander” through The molecules “wander” through the liquid in a random fashionthe liquid in a random fashion

The intermolecular forces are The intermolecular forces are not strong enough to keep the not strong enough to keep the molecules in a fixed positionmolecules in a fixed position

States of MatterStates of Matter

►SolidSolid►LiquidLiquid►GasGas

►PlasmaPlasma

Has Has nono definite volume definite volume

Has Has nono definite shape definite shape

Molecules are in constant random motionMolecules are in constant random motion

The molecules exert only The molecules exert only weak forces on each weak forces on each otherother

Average distance between molecules is large Average distance between molecules is large compared to the size of the moleculescompared to the size of the molecules

States of MatterStates of Matter

►SolidSolid►LiquidLiquid►GasGas►PlasmaPlasma

Matter heated to a Matter heated to a very high temperaturevery high temperature

Many of the electrons are freed from the Many of the electrons are freed from the nucleusnucleus

Result is a collection of free, electrically Result is a collection of free, electrically charged ionscharged ions

Plasmas exist inside stars or experimental Plasmas exist inside stars or experimental reactors or fluorescent light bulbs!reactors or fluorescent light bulbs!For more information:

http://fusedweb.pppl.gov/CPEP/Chart_Pages/4.CreatingConditions.html

Is there a concept that helps to Is there a concept that helps to distinguish between those states of distinguish between those states of

matter?matter?

DensityDensity

► The density of a substance of uniform composition The density of a substance of uniform composition is defined as its mass per unit volume:is defined as its mass per unit volume:

some examples:some examples:

► The densities of most liquids and solids vary The densities of most liquids and solids vary slightlyslightly with changes in temperature and pressure with changes in temperature and pressure

► Densities of gases vary Densities of gases vary greatlygreatly with changes in with changes in temperature and pressure (and generally 1000 temperature and pressure (and generally 1000 smaller)smaller)

V

m

UnitsUnits

SISI kg/mkg/m33

CGSCGS g/cmg/cm3 3 ((1 g/cm1 g/cm33=1000 kg/m=1000 kg/m33 ))

3

2

3

3

4

aV

hRV

RV

cube

cylinder

sphere

Sometimes: Specific GravitySometimes: Specific Gravity

►The The specific gravityspecific gravity of a of a substance is the ratio of its substance is the ratio of its density to the density of water density to the density of water at 4at 4° C° C

The density of water at 4The density of water at 4° C is 1000 ° C is 1000 kg/mkg/m33

►Specific gravity is a Specific gravity is a unitless unitless ratioratio

PressurePressure

► PressurePressure of fluid is the of fluid is the ratio of the ratio of the forceforce exerted exerted by a fluid on a by a fluid on a submerged object to areasubmerged object to area

A

FP

UnitsUnits

SISI Pascal (Pa=N/mPascal (Pa=N/m22))

Example: 100 N over 1 m2 is P=(100 N)/(1 m2)=100 N/m2=100 Pa.

Pressure and Pressure and DepthDepth

► If a fluid is at rest in a container, If a fluid is at rest in a container, all portions of the fluid must be all portions of the fluid must be in static equilibriumin static equilibrium

► All points at the same depth All points at the same depth must be at the must be at the same pressuresame pressure (otherwise, the fluid would not (otherwise, the fluid would not be in equilibrium)be in equilibrium)

► ThreeThree external forces act on the external forces act on the region of a cross-sectional area region of a cross-sectional area AA

External forces: atmospheric, weight, normalExternal forces: atmospheric, weight, normal

AghAPPAAhVM

APMgPAF

0

0

:so,:but

,00ghPP 0

ConcepTest 1ConcepTest 1

You are measuring the pressure at the depth of 10 cm in three different containers. Rank the values of pressure from the greatest to the smallest:

1. 1-2-32. 2-1-33. 3-2-14. It’s the same in all three

Please fill your answer as question 1 of General Purpose Answer Sheet

10 cm

1 2 3

Pressure and Depth equationPressure and Depth equation

► PPoo is normal is normal atmospheric atmospheric pressurepressure 1.013 x 101.013 x 1055 Pa = Pa =

14.7 lb/in14.7 lb/in22

► The pressure does The pressure does not depend upon not depend upon the shape of the the shape of the containercontainer

ghPP o

Other units of pressure: Other units of pressure:

76.0 cm of 76.0 cm of mercurymercury

One atmosphere 1 atm =One atmosphere 1 atm = 1.013 x 101.013 x 1055 Pa Pa

14.7 lb/in14.7 lb/in22

Example:Example:

Given:

masses: h=100 m

Find:

P = ?

Find pressure at 100 m below ocean surface.

pressurecatmospheri1010

1008.910108.9

so,

6

2335

0 2

Pa

msmmkgPaP

ghPP OH

Pascal’s PrinciplePascal’s Principle

► A change in pressure applied A change in pressure applied to an enclosed fluid is to an enclosed fluid is transmitted undiminished to transmitted undiminished to every point of the fluid and every point of the fluid and to the walls of the container.to the walls of the container.

► The hydraulic press is an The hydraulic press is an important application of important application of Pascal’s PrinciplePascal’s Principle

► Also used in hydraulic Also used in hydraulic brakes, forklifts, car lifts, etc.brakes, forklifts, car lifts, etc.

2

2

1

1

A

F

A

FP

Since Since AA22>A>A11, then , then FF22>F>F11 !!!!!!

Measuring PressureMeasuring Pressure

► The spring is calibrated The spring is calibrated by a known forceby a known force

► The force the fluid exerts The force the fluid exerts on the piston is then on the piston is then measuredmeasured

One end of the U-shaped One end of the U-shaped tube is open to the tube is open to the atmosphereatmosphere

The other end is connected The other end is connected to the pressure to be to the pressure to be measuredmeasured

Pressure at B is PPressure at B is Poo++ρghρgh

A long closed tube is A long closed tube is filled with mercury filled with mercury and inverted in a dish and inverted in a dish of mercuryof mercury

Measures Measures atmospheric pressure atmospheric pressure as as ρghρgh

How would you measure How would you measure blood blood pressurepressure??

Has to be: (a) accurate(b) non-invasive(c) simple

sphygmomanometer

QuestionQuestion

Suppose that you placed an extended object in the water. How does the pressure at the top of this object relate to the pressure at the bottom?

1. It’s the same.2. The pressure is greater at the top.3. The pressure is greater at the

bottom.4. Whatever…

Buoyant ForceBuoyant Force

This force is calledThis force is called the buoyant the buoyant force. force.

What is the magnitude of that What is the magnitude of that force?force?

P1A

P2A

mg

!

:,

:but,

11

12

12

gVghAAPghPB

soghPP

APPBF

fluidfluid

Buoyant ForceBuoyant Force

► The magnitude of the buoyant force always The magnitude of the buoyant force always equals the weight of the displaced fluidequals the weight of the displaced fluid

► The buoyant force is the same for a totally The buoyant force is the same for a totally submerged object of any size, shape, or submerged object of any size, shape, or densitydensity

► The buoyant force is The buoyant force is exerted by the fluidexerted by the fluid► Whether an object sinks or floats depends on Whether an object sinks or floats depends on

the relationship between the buoyant force the relationship between the buoyant force and the weightand the weight

fluidfluid wVgB

Archimedes' PrincipleArchimedes' Principle

Any object completely or partially Any object completely or partially submerged in a fluid is buoyed up by a submerged in a fluid is buoyed up by a force whose magnitude is equal to the force whose magnitude is equal to the weight of the fluid displaced by the weight of the fluid displaced by the object.object.

This force is buoyant force.buoyant force.Physical cause: Physical cause: pressure differencepressure difference between the between the toptop and the and the bottombottom of the object of the object

Archimedes’ Principle:Archimedes’ Principle:Totally Submerged ObjectTotally Submerged Object

► The The upward upward buoyant force is buoyant force is B=B=ρρfluidfluidgVgVobjobj

► The The downwarddownward gravitational force is gravitational force is w=mg=ρw=mg=ρobjobjgVgVobjobj

► The The net forcenet force is is B-w=(ρB-w=(ρfluidfluid-ρ-ρobjobj)gV)gVobjobj

Depending on the Depending on the direction of the net direction of the net force, the object will force, the object will either either float upfloat up or or sinksink!!

► The object is The object is less denseless dense than the fluid than the fluid ρρfluidfluid<ρ<ρobjobj

► The object experiences a The object experiences a net net upwardupward force force

The The net forcenet force is is B-w=(ρB-w=(ρfluidfluid-ρ-ρobjobj)gV)gVobjobj

The object is The object is more more densedense than the fluid than the fluid ρρfluidfluid>ρ>ρobjobj

The net force is The net force is downwarddownward, so the object , so the object accelerates downwardaccelerates downward

Archimedes’ Principle:Archimedes’ Principle:Floating ObjectFloating Object

► The object is in The object is in static equilibriumstatic equilibrium► The upward buoyant force is The upward buoyant force is

balanced by the downward force balanced by the downward force of gravityof gravity

► Volume of the fluid displaced Volume of the fluid displaced corresponds to the volume of the corresponds to the volume of the object beneath the fluid levelobject beneath the fluid level

obj

fluid

fluid

obj

V

V

or,:If objectobjectfluidfluid gVgVmgB

Question 1Question 1

Suppose that you have a steel bar. Will it float on water? Why?

Question 2Question 2

Suppose that you have a steel bar. Will it float on water? Why?

How come that ships (which are made of steel) can float?

Question 3Question 3

Suppose that your friend gave you a necklace (crown, piece of yellow metal, …). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)?

Question 3Question 3

Suppose that your friend gave you a necklace (crown, piece of yellow metal, …). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)?

Idea: determine density! Let’s weight the object in and outside the water container:

gV

mg

orgVmg

g

mgweightV

gVmg

Bmgweight

fluid

fluid

""

""

If is not the same as gold, your friend is lying…

Out:Out: In:In:

ConcepTest 2ConcepTest 2

Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.Which weighs more?

1. The glass without ice cubes.2. The glass with ice cubes.3. The two weigh the same.

Please fill your answer as question 3 of General Purpose Answer Sheet

ConcepTest 3ConcepTest 3

Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.When the ice cubes melt, in which glass is the level of the waterhigher?

1. The glass without ice cubes.2. The glass with ice cubes.3. It is the same in both.

Please fill your answer as question 5 of General Purpose Answer Sheet

Fluids in Motion:Fluids in Motion:Streamline FlowStreamline Flow

► Streamline flow Streamline flow every particle that passes a particular point every particle that passes a particular point

moves exactly along the smooth path moves exactly along the smooth path followed by particles that passed the point followed by particles that passed the point earlierearlier

also called laminar flowalso called laminar flow

► Streamline is the pathStreamline is the path different streamlines cannot cross each different streamlines cannot cross each

otherother the streamline at any point coincides with the streamline at any point coincides with

the direction of fluid velocity at that pointthe direction of fluid velocity at that point

Fluids in Motion:Fluids in Motion:Turbulent FlowTurbulent Flow

►The flow becomes irregularThe flow becomes irregular exceeds a certain velocityexceeds a certain velocity any condition that causes abrupt changes any condition that causes abrupt changes

in velocityin velocity

►Eddy currents are a characteristic of Eddy currents are a characteristic of turbulent flowturbulent flow

Fluid Flow: ViscosityFluid Flow: Viscosity

►Viscosity is the degree of internal Viscosity is the degree of internal friction in the fluidfriction in the fluid

►The internal friction is associated with The internal friction is associated with the resistance between two adjacent the resistance between two adjacent layers of the fluid moving relative to layers of the fluid moving relative to each othereach other

Characteristics of an Ideal Characteristics of an Ideal FluidFluid

► The fluid is The fluid is nonviscousnonviscous There is no internal friction between adjacent There is no internal friction between adjacent

layerslayers► The fluid is The fluid is incompressibleincompressible

Its density is constantIts density is constant► The fluid is The fluid is steadysteady

Its velocity, density and pressure do not change Its velocity, density and pressure do not change in timein time

► The fluid The fluid moves without turbulencemoves without turbulence No eddy currents are presentNo eddy currents are present

Equation of ContinuityEquation of Continuity

► AA11vv11 = A = A22vv22

► The product of the The product of the cross-sectional area cross-sectional area of a pipe and the fluid of a pipe and the fluid speed is a constantspeed is a constant Speed is high where Speed is high where

the pipe is narrow and the pipe is narrow and speed is low where the speed is low where the pipe has a large pipe has a large diameterdiameter

► Av is called the Av is called the flow flow raterate

Bernoulli’s EquationBernoulli’s Equation

► Relates pressure to fluid speed and elevationRelates pressure to fluid speed and elevation► Bernoulli’s equation is a consequence of Bernoulli’s equation is a consequence of

Conservation of Energy applied to an ideal fluidConservation of Energy applied to an ideal fluid► Assumes the fluid is incompressible and Assumes the fluid is incompressible and

nonviscous, and flows in a nonturbulent, steady-nonviscous, and flows in a nonturbulent, steady-state mannerstate manner

► States that the sum of the pressure, kinetic States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume, and the potential energy per unit volume has the same value at energy per unit volume has the same value at all points along a streamlineall points along a streamline

constant gyvP 2

2

1

How to measure the speed of the How to measure the speed of the fluid flow: Venturi Meterfluid flow: Venturi Meter

► Shows fluid flowing Shows fluid flowing through a horizontal through a horizontal constricted pipeconstricted pipe

► Speed changes as Speed changes as diameter changesdiameter changes

► Swiftly moving fluids Swiftly moving fluids exert less pressure exert less pressure than do slowly moving than do slowly moving fluidsfluids