Archimedes’ Principle

Physics 202Professor Lee

CarknerLecture 2

“Got to write a book, see, to prove you’re a philosopher. Then you get your … free official philosopher’s loofah.”

--Terry Pratchett, Small Gods

PAL #1 Fluids Column of water to produce 1 atm of

pressure

= 1000 kg/m3

h = P/g = 10.3 m Double diameter, pressure does not change

On Mars pressure would decrease

Archimedes’ Principle The fluid exerts a force on the object

If you measure the buoyant force and the

weight of the displaced fluid, you find: An object in a fluid is supported by a buoyant

force equal to the weight of fluid it displaces Applies to objects both floating and

submerged

Buoyancy

Will it Float?

What determines if a object will sink or float?

A floating object displaces fluid equal

to its weight

A sinking object displaces fluid equal

to its volume

Floating How will an object float? The denser the object, the lower it will float, or:

Example: ice floating in water,

W=Vg

Vi/Vw=w/i

w = 1024 kg/m3 and i = 917 kg/m3

Iceberg

Ideal Fluids

Steady --

Incompressible -- Nonviscous -- Irrotational --

Real fluids are much more complicated The ideal fluid approximation is usually not

very good

Moving Fluids Consider a pipe of cross sectional area A with a fluid

moving through it with velocity v

Mass must be conserved so,

If the density is constant then,Av= constant = R = volume flow rate

Because the amount of fluid going in must equal the

amount of fluid going out

Continuity

R=Av=constant is called the equation of continuity

You can use it to determine the flow rates of a system of pipes

Can’t lose or gain any material

Continuity

The Prancing Fluids

How can we keep track of it all? The laws of physics must be obeyed

Neither energy nor matter can be

created or destroyed

Bernoulli’s Equation Consider a pipe that bends up and gets wider at

the far end with fluid being forced through it The work of the system due to lifting the fluid is,

The work of the system due to pressure is,

Wp=Fd=pAd=pV=-(p2-p1)V The change in kinetic energy is,

Equating work and KE yields,

p1+(1/2)v12+gy1=p2+(1/2)v2

2+gy2

Fluid Flow

Consequences of Bernoulli’s Equation

Fast moving fluids exert less pressure than slow moving fluids

This is known as Bernoulli’s principle Based on conservation of energy

Note that Bernoulli only holds for moving fluids

Constricted Flow

Bernoulli in Action

Blowing between two pieces of paper

Convertible top bulging out Shower curtains getting sucked

into the shower

Shower Physics

Lift

Consider a thin surface with air flowing above and below it

This force is called lift If you can somehow get air to flow over

an object to produce lift, what happens?

December 17, 1903

Deriving Lift Consider a wing of area A, in air of density Use Bernoulli’s equation:

The difference in pressure is:

pb-pt=1/2vt2-1/2vb

2

Pressure is F/A so:

L=Fb-Ft and so:

If the lift is greater than the weight of the plane, you fly

Summary: Fluid Basics Density ==m/V Pressure=p=F/A On Earth the atmosphere exerts a

pressure and gravity causes columns of fluid to exert pressure

Pressure of column of fluid:p=p0+gh

For fluid of uniform density, pressure only depends on height

Summary: Pascal and Archimedes

Pascal -- pressure on one part of fluid is transmitted to every other part

Hydraulic lever -- A small force applied for a large distance can be transformed into a large force over a short distance

Fo=Fi(Ao/Ai) and do=di(Ai/Ao) Archimedes -- An object is buoyed up by

a force equal to the weight of the fluid it displaces Must be less dense than fluid to float

Summary: Moving Fluids

Continuity -- the volume flow rate (R=Av) is a constant fluid moving into a narrower pipe

speeds up Bernoullip1+1/2v1

2+gy1=p2+1/2v22+gy2

Slow moving fluids exert more pressure than fast moving fluids