Liquids & Buoyant Force

Notes (p275 HRW)

Liquids

Study of liquids

• Hydrostatics – liquids that are stationary

• Archimedes, Pascal

• Hydrodynamics – liquids that are moving

• Bernouilli

Fundamental Characteristic

Density ()

How is density defined? • mass/volume

• = m/V

What are density units in SI • kg/m3 (standard) or gm/cc (common)

Example • water = 1000 kg/m3 (or 1 gm/cc)

• gold = 19.3 x water

• lead = 11.3 x

• iron = 7.8 x

• ice = 0.92 x

• balsa = 0.12 x

• air = 1.2 kg/m3

Lower density floats

on higher density

Pressure (p)

Definition:

• force perpendicular to plane/area over which force is acting

• P = Force/Area = F/A

• SI Units = pascals (Newtons/meter2)

• Air Pressure

• Standard air pressure @ sea level and 20C, expressed as:

• 1.01 x 105 Pa

• 1 atmosphere

• 760 mm (or 76 cm) of mercury

• 30 inches of mercury

• Measured by a barometer or manometer

• Average tire pressure ~ 200 kPa or 30 psi (lbs/in2)

Hydrostatic (Gauge) Pressure

Fluid pressure increases with depth because the water at a depth must support the weight of water above it. • Ex.

• Diving to the bottom of the deep end of a pool – what do you feel?

• Air pressure

P = ρgh, where • ρ = density of fluid

• g = 9.8 m/sec2

• h = depth of fluid (ex. height of water column)

• This is gauge pressure

Container shape?

NO EFFECT!

Absolute/Total Pressure

Add atmospheric pressure (Pair) which acts on the surface of the fluid and the total pressure becomes • Ptotal = Pair + ρgh

Generically • The pressure at the bottom of a

column (Pb) equals the pressure at the top (Pt) PLUS the pressure due to the column, or

• Pb = Pt + ρgh

Practice - submarines

A sub dives to a depth of 200 m. How

much water pressure must the hull be

able to withstand, or what is the (gauge)

water pressure at 200 m?

• Solve P = ρgh

• P = 1000 x 9.8 x 200

• P = 1.96 x 106 Pa

Every sq meter of surface must

withstand ~ 2 million newtons

Note: in this example we have NOT included the air pressure pushing on the top of the water!

Submarines

Maximum depth – classified, but

generally believed to be ~1500 ft for US

and ~2500 ft for Russian

Sub disasters –

• Thresher (‘63), sank off Cape Cod in 8400 ft

after joint to outer hull failed, flooding sub

• Kursk (2000), Barents Sea, 350 ft after

torpedo accident

Practice

How deep (h) must a diver go before he experiences another atmosphere (1.01 x 105 Pa) of water pressure?

Solve for h in water (ρ = 1000 kg/m3) • P = ρgh

• 1.01 x 105 = 1000 x 9.8 x h

• h = 10.31 m (~34 feet)

Roughly, every 30 feet of dive adds 1 atmosphere of pressure to a diver

Interesting facts on diving!

Deepest free dive

• 124 m (~400 ft) feet

Deepest assisted dive

• 214 m (~700 ft)

Breath holding record

• >19 minutes

Mini-lab

Density determination of 3 samples

• Measure

• Write-up

• Submit

Pascal’s Principle

Pressure applied to an enclosed fluid is

transmitted equally and undiminished to

every part of the fluid, as well as the walls

of the container.

• P1 = P2, (Pressure a point 1 = pressure at point 2)

2

2

1

1

A

F

A

F

Application of Pascal’s Principle

Hydraulic lift – service stations

Examples of Pascal’s Principle

Examples

Example

The small piston of a hydraulic lift has an

area of 0.2 m2. If a car weighing 1.2e4 N

sits on the large piston, area 0.9 m2, how

large a force must be applied to the small

piston to support the car?

F1/A1 = F2/A2

• F1/0.2 = 1.2e4/0.9

• F1 = 0.2*1.2e4/0.9 = 2.7e3 N

What is buoyancy?

In physics, buoyancy is

the upward force acting

on an object in a fluid,

and can be:

• Positive

• Neutral

• Negative

Alligators & crocodiles?

What causes buoyancy?

Buoyancy is the result of the difference

in pressure exerted on the top and

bottom of an object.

Question

If you place a rock into a

bucket of water, say ½ filled,

what will you observe about

the water line in the bucket?

It will rise to reflect the

volume of the rock – if the

rock was 0.1 m3, the water

would rise 0.1 m3, b/c the

rock is submerged.

This is an example of Archimedes’ Principle…

Archimedes Principle - example

In air the stone

weighs 44 N

In water the stone

weighs 36 N

Buoyant force is the

difference = 8N ↑

Buoyancy Example

Weight in

air Apparent

weight

Fb = Weight in air – weight in water

Buoyancy - Archimedes

Archimedes’ Principle • When a body is partly or wholly submerged in a fluid,

it will experience a buoyant force (Fb)equal to the weight of the fluid displaced.

• Buoyant (Upthrust) force (Fb) • Fb = Weightfluid displaced = (massfluid)*(g)

• Fb = (fluid)*(Vfluid displaced)*(g) (Note: =m/V or m= *V)

• Fb = (V)*g directed upward!

• Case 1 – object is submerged • Fb = (fluid)*(Vfluid displaced)*(g)

• Case 2 – object is partially submerged (floating) • Fb = (object)*(Vobject)*(g)

Buoyancy Summary Table

IF THEN

ρobject > ρfluid Wobj↓ > Fb ↑ sinking

ρobject = ρfluid Wobj↓ = Fb ↑ neither

sink nor float

ρobject < ρfluid Wobj↓ < Fb ↑ float

ρ = density, W = weight Fb = buoyant force

Example of Archimedes

Principle & Buoyant Force

If a house brick of density (ρ) 2000 kg/m3

and volume (V) of 0.00123 m3 is placed in

a bucket of water, what is the buoyant force

(Fb) acting on a brick?

Solve: Fb = ρVg

Fb = 1000 x 0.00123 x 9.8

Fb = 12.05 N

Buoyant Force Practice

An ice cube is floating in a glass of water

(ρ=1000 kg/m3). The ice, whose density is

917 kg/m3, has dimensions of 0.03 x 0.02 x

0.02 m. What is the buoyant force on the ice?

Solve: Note: the ice is floating thus the

buoyant force↑ = weight of the ice↓

• Fb = (ρV)*g (use ρ, V for ice, not water)

• Fb = 917*(0.03*0.02*0.02)*9.8

• Fb = 0.11 N↑

Buoyant Force Practice

A ferry boat is 4 m wide and 6 m long. When a truck pulls onto it, the boat sinks 4 cm in the water. • What is the weight of the truck?

Use Archimedes Principle: • Weight of truck = weight of water displaced (find

this displacement)

• Weight = mass x g • Mass of water (m) = density x volume

• m= 1000 x (4 x 6 x 0.04) = 960 kg

• Weight = m x g = 960 x 9.8 = 9,408N

Buoyant Force Practice

Example

• A piece of metal weighs 7.84N in air and

6.86N when completely immersed in water.

• What is the buoyant force?

• Fb = 7.84 – 6.86 = 0.98N

• What is the density (ρ) of the object?

• Fb = ρVg (ρ, V refer to the fluid when submerged)

• V = Fb /ρg = 0.98/(1000*9.8) = 0.0001 m3

• Mass = Weight/9.8 = 0.8 kg

• ρ = m/V = 0.8/0.0001 = 8000 kg/m3

Buoyancy - icebergs

If an object is floating on the

surface, then the volume that

is below the surface can be

determined as follows:

• Vf/Vo = o/f, where

• Vf= volume of object that is below

surface

• Vo= volume of the object

• o = density of the object

• f = density of the fluid

% Iceberg in salt water that is below surface = .92/1.025 = 89%