23-Sep-10

Mass Density(Chapter 12)

PressureBuoyant Force

(Chap. 13)

Lecture 13

DensityThe (mass) density of a material is its mass M per unit volume V:

The densities of most liquids and solids vary The densities of most liquids and solids vary slightly with changes in temperature and slightly with changes in temperature and pressurepressure

Densities of gases vary greatly with changes Densities of gases vary greatly with changes in temperature and pressurein temperature and pressure

Density ValuesMaterial Density (kg/m3)Water (at 4°C) 1,000 (1 g/cm3)Lead 11,300Gold 19,300Air 1.29Helium 0.18The specific gravity of a substance is the

ratio of its density to that of water.

23-Sep-10 Physics 101

Demo: Gold Brick

Compute density of gold brick.Dimensions: (20.5 cm)x(10 cm)x(5 cm)Volume: 1025 cubic centimeters (1.025 x 10-3 m3)Mass: 19.8 kgDensity = (19.8 kg) / (1.025 x 10-3) = 19.3 kg/m3

Check with table that this is density of gold

23-Sep-10 Physics 101

Demo: Fool’s GoldUse displacement to find irregular volumes.Pyrite (called Fool’s Gold) is an iron ore (FeS2).Pyrite sample has:Mass: 0.061 kgVolume: 12 x 10-6 m3

Density=(.061)/(12x10-6) )= 5.1x 103 kg/m3

About a quarter the density of real gold.

23-Sep-10 Physics 101

Check YourselfWater expands when it freezes into ice.

Which has more volume, kilogram of ice or kg of water?

Which has more mass, a kg of ice or a kg of water?

Which has a higher density, water or ice (or the same)?

Which has more mass, a liter of ice or a liter of water?

PressurePressure is force per unit area:

1 Pa = 1 N/m2

Other common pressure units:Pounds per Square inch (PSI) - tires, etc.mm Hg - blood pressureinches Hg - weather barometer

or Pascal (Pa)

Pressure vs. Contact AreaThe same force applied over a smaller area results in greater pressure – think of poking a balloon with your finger and then with a needle.

Example• What is pressure exerted by 80-kg person’s shoe

on floor if shoe is 0.3 m by 0.1 m and each shoe supports half of total weight?

• What if half of weight rests on 0.005m by 0.005m stiletto heel?

• P= 1.6 x 107 Pa = 16,000 kPa

24 /103.1)1.0)(3.0(

2/)/8.9)(80(2/ mNxmmkgNkg

AWP ===

=13 kPa

23-Sep-10

Sample Problem

“Gold” brick is roughly:12 kilograms [Mass]120 Newtons [Weight](0.2 m)x(0.1 m) =

(0.02 m2) [Area]Pressure on surface is(120)/(0.02)

= 6000 Pascals

Note: Atmospheric pressure is about 100,000 Pascals so much more than pressure due to the “gold” brick.

23-Sep-10

Check YourselfIn which case is the pressure greatest? A B

A B

23-Sep-10

Check YourselfIn which case is the pressure greatest? A B

AB

23-Sep-10

Demo: Bed of NailsOne may safely lie or sit on

a bed of nails, as long as there are enough nails so that the pressure, measured as force per nail, is small.

Weight of 150 pounds is distributed over 300 nails. Force per nail is ½ lb. Need 5 lb per nail to pierce skin.

How to Measure Blood Pressure?Has to be: (a) accurate

(b) non-invasive(c) simple

Sphygmo-manometer

Blood Pressure Readings

• The higher (systolic) number represents pressure while the heart contracts to pump blood to the body.

• The lower (diastolic) number represents pressure when the heart relaxes between beats.

• The systolic pressure is always stated first. For example: 118/76 (118 over 76); systolic = 118, diastolic = 76. Blood pressure below 120 over 80 mmHg (millimeters of mercury) is considered optimal for adults.

Pressure in Fluids

Pressure is the same in every direction in a fluid at a given depth; if it were not, the fluid would flow.

23-Sep-10

Pressure in Liquids

Pressure in a liquid depends on depth.As with bricks, weight of what’s above

determines pressure.

High

Medium

Low

High

Medium

Low

Pressure versus Depth in FluidThere is an increase in fluid pressure as we

descend further below top surface. This is due to increasing weight of fluid above us.

For an open container of cross-sectional area A and height h filled with fluid of density ρ

Fbottom = PatA + ρAhg

Pbottom = Fbottom/A

Pressure vs. DepthThe same relation works between any two points separated by vertical distance h.

This relation is valid for any liquid whose density does not change with depth.

Example: Find pressure at 100 m below ocean surface.

P = Pat+ρWgh

P = 101 kPa+(103 kg/m3)(9.8 N/kg)(100m)

= 1081 kPa (about 10 times Pat)

Pressure and DepthPressure-depth relation is true in any container where the fluid can flow freely – the pressure at the same height will be the same everywhere.

“Fluid Seeks It’s Own Level”

Question

• Three containers a), b), c) each have the same bottom surface area, A, and are each filled with water to the same height h.

• Pressure as a function of depth = Patm + ρgh.

Which container has the greatest pressure of water pushing on the bottom surface?

a) b) c)

23-Sep-10

BuoyancySince pressure

depends on depth, a submerged object has more force due to pressure below it than above it.

Net effect is to have a net upward force, which we call buoyancy.

Buoyancy

Weight

If weight exceeds buoyancy force then object sinks, otherwise it floats.

Pressure Pressure

Pressure

23-Sep-10

Archimedes’ PrincipleWeight of liquid displaced by floating or submerged

object equals the buoyant force on the object.

Archimedes’ PrincipleObject completely immersed in a fluid experiences upward buoyant force equal in magnitude to weight of fluid displaced by object.

•Buoyant force is the same for a totally submerged object of any size, shape, or density•The buoyant force is exerted by the fluid•Whether an object sinks or floats depends on relationship between buoyant force and weight

f

FB=ρfVg

23-Sep-10

Demo: Take a Dip

A 1 kg mass submerged in water displaces 150 cm3

(1.5 x 10-4 m3).Water has density of 1000

kg/m3 so displaced water has mass of 0.15 kg.

Buoyant force equals weight of 0.15 kg(150 g).

1000 g

1000 g

850 g

150 cc displaced

Beaker ofWater

23-Sep-10

Demo: Floating in the Tub Buoyant force depends only on the volume

of fluid displaced, not on the volume of fluid in which the object is immersed.

Rubber duck floats the same in a little water or in a full beaker of water

23-Sep-10

Check Yourself

?

Place block of wood in the water. Scale reading goes up, down, or stays the same?

50 N

10 N Block

10 N Water

40 N Water

23-Sep-10

Check YourselfUse similar block of petrified wood. Scale reading goes up, down, or stays the same?

?

50 N

30 N Block

10 N Water

40 N Water

Applications of Archimedes’ Principle

An object floatswhen it displaces an amount of fluid equal to its weight.

Some fraction of object volume will be submerged.

23-Sep-10

Density & Floating

By Archimedes’principle, a solid object will float if the density of the object is less than the density of the liquid.

Billiard ball (4.0 g/cm3) floating in a cup of mercury (13.6 g/cm3)

23-Sep-10

Demo: Diet or Regular?What can you say

about the density of diet cola as compared with regular cola?

23-Sep-10

ShipsDensity of wood is

about 0.5 to 0.8 g/cm3

so not surprising that wooden ships float.

Density of iron 7.9 g/cm3. How is it that a battleship can float?

23-Sep-10

Density & Floating, RevisitedWhen an object is not solid then it floats if the

average density, (total mass)/(total volume), is less than the density of the liquid.

Solid iron ballVolume: 100 ccMass: 790 gDensity: 7.9 g/cc

Iron ball, 90% hollowVolume: 1000 ccMass: 790 gDensity: 0.79 g/cc

23-Sep-10

Iron Ships, Floating & Sinking

Iron ship floats since it is hollow inside. If water floods the inside then it sinks.

23-Sep-10

Floating & Liquid Density

The greater the density of a liquid, the greater the buoyant force on objects floating or immersed in the liquid. Floating in Great Salt Lake,

Utah is easy because the lake water is dense due to high concentration of salt.

Iceberg: ρ(ice)<ρ(water)An Iceberg is floating in the ocean. As the iceberg melts, does the ocean level 1) rise, 2) sink, or 3) stay the same.

h

As ice, the iceberg displaces a volume of water equal in mass to the iceberg’s mass. Once it is melted, the iceberg still displaces a volume of water equal in mass to the original iceberg (melting doesn’t change the iceberg mass).

23-Sep-10

Demo: Hydrometer

Hydrometer is a calibrated float; denser the liquid, the higher the hydrometer floats.

23-Sep-10

Density & Winemaking“Take a look at Diagram A, on the left, which represents a test jar full of must before the yeast is pitched.

Do you see how the hydrometer is floating rather high? This is because the liquid is "heavy" with all the sugar... the hydrometer is pushed up because of this.

As the yeast turns the sugar into alcohol and carbon dioxide, the wine becomes lighter (alcohol weighs less per unit volume) and the hydrometer doesn't float as high as it once did. Diagram B (seen on the right) represents a wine that has fermented to dryness and is lighter than water.”

www.grapestompers.com

A B

Key Points of Lecture 13Key Points of Lecture 13

Before next lecture, read Hewitt Chap. 13.

Homework Assignment #9 is due before 11:00 PM on Sunday, Sept. 26.

Homework Assignment #10 is due before 11:00 PM on Tuesday, Sept. 28.

• Mass Density• Pressure• Buoyant Force• Archimedes’ Principle