Buoyancy and Archimedes Principle

Lecturer: Professor Stephen T. Thornton

Reading QuizReading Quiz

A) diminish

B) stop altogether

C) go out in a straight line

D) curve upwards

When a hole is made in the side of a When a hole is made in the side of a

Coke can holding water, water flows Coke can holding water, water flows

out and follows a parabolic trajectory. out and follows a parabolic trajectory.

If the container is dropped in free If the container is dropped in free

fall, the water flow will:fall, the water flow will:

Co

ca-C

ola

Reading QuizReading Quiz

Water flows out of the hole because the

water pressure insidewater pressure inside is larger than the air air

pressure outsidepressure outside. The water pressure is

due to the weightweight of the water. When the

can is in free fall, the water is weightlessweightless,

so the water pressure is zero, and hence

no water is pushed out of the hole!

A) diminish

B) stop altogether

C) go out in a straight line

D) curve upwards

When a hole is made in the side of a When a hole is made in the side of a

Coke can holding water, water flows Coke can holding water, water flows

out and follows a parabolic trajectory. out and follows a parabolic trajectory.

If the container is dropped in free If the container is dropped in free

fall, the water flow will:fall, the water flow will:

Co

ca-C

ola

Last Time

Density, fluids

Pressure

Pressure gauges and barometers

TodayMore on pressurePascal’s PrincipleBuoyancyArchimedes PrincipleMass flow – equation of continuity

Today we are studying fluid dynamics after Archimedes Principle.

There are a number of different types of pressure gauges. This one is an open-tube manometer. The pressure in the open end is atmospheric pressure; the pressure being measured will cause the fluid to rise until the pressures on both sides at the same height are equal.

Measurement of Pressure; Gauges and the Barometer

Tire pressure gauge.

Pascal’s Principle:

An external pressure applied to an enclosed fluid is transmitted unchanged to every point within the fluid.

Do Pascal’s vases demoDo Cartesian diver demo

A Hydraulic Lift

1 2

1 2

F FP

A A

In the hydraulic lift (show demo at end), the liquid is enclosed, and the pressure is the same throughout.

1 2

1 2

22 1

1

1 1 2 2

12 1 2 1

2

large increase in force

But volumes are equal,

so

F FP

A A

AF F

A

V A d A d

Ad d d d

A

Pascal’s Principle

Conceptual QuizConceptual Quiz

A) container 1

B) container 2

C) container 3

D) all three are equal

Three open containers are filled with water to the Three open containers are filled with water to the

same height and have the same surface area at same height and have the same surface area at

the base, but the total weight of water is different the base, but the total weight of water is different

for each. Which container has the greatest total for each. Which container has the greatest total

force acting on its base?force acting on its base?

Conceptual QuizConceptual Quiz

The pressure at the bottom of each

container depends only on the height

of water above it! This is the same for

all the containers. The total force is

the product of the pressure times the

area of the base, but since the base is

also the same for all containers, the

total force is the same.

A) container 1

B) container 2

C) container 3

D) all three are equal

Three open containers are filled with water to the Three open containers are filled with water to the

same height and have the same surface area at same height and have the same surface area at

the base, but the total weight of water is different the base, but the total weight of water is different

for each. Which container has the greatest total for each. Which container has the greatest total

force acting on its base?force acting on its base?

When you drink liquid through a straw, which of the items listed is primarily responsible for this to function?

A) water pressure

B) gravity

C) inertia

D) atmospheric pressure

E) mass

Conceptual QuizConceptual Quiz

When you drink liquid through a straw, which of the items listed below is primarily responsible for this to function?

A) water pressure

B) gravity

C) inertia

D) atmospheric pressure

E) mass

When you suck on a straw, you expand your lungs, which reduces the air pressure inside your mouth to less than atmospheric pressure. Then the atmospheric pressure pushing on the liquid in the glass provides a net upward force on the liquid in the straw sufficient to push the liquid up the straw.

Conceptual QuizConceptual Quiz

Follow-up:Follow-up: Is it possible to sip liquid through a straw on the Moon?

A) greater than PA

B) equal to PA

C) less than PA

You put a straw into a glass of water, place You put a straw into a glass of water, place

your finger over the top so no air can get in your finger over the top so no air can get in

or out, and then lift the straw from the or out, and then lift the straw from the

liquid. You find that the straw retains some liquid. You find that the straw retains some

liquid. How does the air pressure liquid. How does the air pressure PP in the in the

upper part compare to atmospheric upper part compare to atmospheric

pressure pressure PPAA??

Conceptual QuizConceptual Quiz

Consider the forces acting at the bottom of the

straw: PPAA – – PP – – g H g H = = 00

This point is in equilibrium, so net force is This point is in equilibrium, so net force is zero.zero.

Thus, PP = = PPAA – – g H g H and so we see that

the pressure pressure PP inside the straw must be lessless

than the outside pressure outside pressure PPAA.

H

You put a straw into a glass of water, place You put a straw into a glass of water, place

your finger over the top so no air can get in your finger over the top so no air can get in

or out, and then lift the straw from the or out, and then lift the straw from the

liquid. You find that the straw retains some liquid. You find that the straw retains some

liquid. How does the air pressure liquid. How does the air pressure PP in the in the

upper part compare to atmospheric upper part compare to atmospheric

pressure pressure PPAA??

A) greater than PA

B) equal to PA

C) less than PA

Conceptual QuizConceptual Quiz

Buoyant Force Due to a Fluid

Pressure is higher below2 1

P P gL

The buoyant force is due to the difference in force at the bottom and the top of the cube of sides L.

2 2 3b 2 1 2 1

b General result

F F F P P L gL L gL

F gV mg

Archimedes’ principle:

An object completely immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object.

Buoyant force equals the weight of the displaced liquid, not the weight of the object!

Buoyancy and Archimedes’ Principle

Archimedes’ principle: The buoyant force on an immersed object equals the weight of displacedfluid.The picture below shows an object in the air, partially submerged, and completely submerged.

Do Archimedes’ Principle I demo.

Start Archimedes’ Principle II demo.

Flotation

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

The total weight of the block equals the weight of the fluid displaced.

Floating an Object That Is More Dense than Water

The wood and water in (a) have the same weight, and the metal block and water in (b) have the same weight.Note in (c) that bowl made of the metal floats, because the bowl is hollow.

sub

ssub s

f

Volume of submerged block = V

V V

Remember icebergs!

volume of solidsV

s sf sub

weight of displaced water = weight of block

BF mgV g V g

is water density

Conceptual Quiz:What happens to the water level when the ice melts?

A) Water overflows.B) Water level decreases.C) Water level stays the same.

Answer: C

Water level stays the same. Because the ice cube is floating, it displaces a volume of water equal to its weight. When it melts it becomes water and displaces the same volume of water it displaced. Remember the density of ice is less than that of water.

Conceptual Quiz: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? A)    The glass without ice cubes.B)    The glass with ice cubes.C)    The two weigh the same.

Answer: C

The ice cubes displace exactly their own weight in water, so the two glasses weigh the same amount. It is essential that the ice sticks out above the level of the water.

Conceptual Quiz:Now a pebble sits on top of the ice, and the water is filled to the brim of the glass. What happens when the ice melts?

A) The water overflows.B) The water level decreases.C) The water level stays the same.D) The pebble explodes.

Answer: B

The ice makes no difference, but now the pebble also displaces an amount of water equal to the pebble’s weight. When the ice melts, the pebble drops to the bottom and displaces a volume of water equal to its own volume. Because the volume of the pebble is less than the volume of the water it displaced originally, there is less water displaced afterwards. The water level drops.

Turbulent flow

Laminar flow, also known as streamline flow.

Consider a mass m passing through a particular point in time t. The ratio m/ t must be constant at every point or mass will build up.

1 1 1 2 2 2

makes sense, because is volume/time.

If mass flow is constant, then

called " "

mAv Av

t

Av A v equation of continuity

Gases are compressible, but liquids are not. For liquid flow, the density is constant. Then for liquids.

Useful equation for many applications: for example, water hoses.

1 1 2 2A v A v=

When using a water hose, we put our thumb over end to increase water speed.Hose nozzle does the same thing. Decrease area, increase speed.

Water nozzle

A1v1 = A2v2

small A, large v

Conceptual QuizConceptual Quiz

1

2

A) greater

B) the same

C) smaller

Imagine holding two identical Imagine holding two identical

bricks in place underwater. Brick bricks in place underwater. Brick

1 is just beneath the surface of 1 is just beneath the surface of

the water, and brick 2 is held the water, and brick 2 is held

about 2 feet down. The force about 2 feet down. The force

needed to hold brick 2 in place is:needed to hold brick 2 in place is:

The force needed to hold the brick in The force needed to hold the brick in

place underwater is place underwater is WW – – FFBB.. According

to Archimedes’ Principle, FB is equal to

the weight of the fluid displaced.

Because each brick displaces the same Because each brick displaces the same

amount of fluid, then amount of fluid, then FFBB is the same in is the same in

both cases.both cases.

Conceptual QuizConceptual Quiz

1

2

A) greater

B) the same

C) smaller

Imagine holding two identical Imagine holding two identical

bricks in place underwater. Brick bricks in place underwater. Brick

1 is just beneath the surface of 1 is just beneath the surface of

the water, and brick 2 is held the water, and brick 2 is held

about 2 feet down. The force about 2 feet down. The force

needed to hold brick 2 in place is:needed to hold brick 2 in place is: