Archimedesí Principle


Physics 202
Professor Lee Carkner
Lecture 2

PAL #1 Fluids

Column of water to produce 1 atm of pressure
P = rgh
P =
r = 1000 kg/m3
g = 9.8 m/s2
h = P/rg =
Double diameter, pressure does not change

On Mars pressure would decrease
Mars has smaller value of g

Archimedesí Principle

What happens if you put an object in a fluid?

Called the buoyant force

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


Will it Float?


Density
An object less dense than the fluid will 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 volume of fluid displaced is proportional to the ratio of the densities
Example: ice floating in water,

riVig=rwVwg

Vw=Vi (ri/rw)
rw = 1024 kg/m3 and ri = 917 kg/m3

Ideal Fluids

Steady --

Incompressible -- density is constant
Nonviscous --
Irrotational -- constant velocity through a cross section

The ideal fluid approximation is usually not very good

Moving Fluids



What happens if the pipe narrows?

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


Constricting a flow increases its velocity
Because the amount of fluid going in must equal the amount of fluid going out
Or, a big slow flow moves as much mass as a small fast flow

Continuity

R=Av=constant is called the equation of continuity

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

Canít lose or gain any material

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

Bernoulliís Equation

Consider a pipe that bends up and gets wider at the far end with fluid being forced through it


Wg = -Dmg(y2-y1) = -rgDV(y2-y1)
The work of the system due to pressure is,
Wp=Fd=pAd=DpDV=-(p2-p1)DV

D(1/2mv2)=1/2rDV(v22-v12)

p1+(1/2)rv12+rgy1=p2+(1/2)rv22+rgy2

Consequences of Bernoulliís



Fast moving fluids exert less pressure than slow moving fluids
This is known as Bernoulliís principle


Energy that goes into velocity cannot go into pressure
Note that Bernoulli only holds for moving fluids

Bernoulli in Action

Blowing between two pieces of paper


Convertible top bulging out
Airplanes taking off into the wind


Lift



If the velocity of the flow is less on the bottom than on top there is a net pressure on the bottom and thus a net force pushing up

If you can somehow get air to flow over an object to produce lift, what happens?

Deriving Lift


Use Bernoulliís equation:
pt+1/2rvt2=pb+1/2rvb2
The difference in pressure is:
pb-pt=1/2rvt2-1/2rvb2

(Fb/A)-(Ft/A)=1/2r(vt2-vb2)

L= (Ĺ)rA(vt2-vb2)


Next Time

Read: 15.1-15.3
Homework: Ch 14, P: 37, 42, 47, Ch 15, P: 6, 7

Which of the following would decrease the pressure you exert on the floor the most?

Doubling your mass
Doubling the mass of the earth
Doubling your height
Doubling the size of your shoes
Doubling air pressure

Which of the following would increase the pressure of a column of fluid of fixed mass the most?

Doubling the width of the column
Halving the density of the fluid
Halving the mass of the Earth
Halving the speed of the Earthís rotation
Doubling the height of the column


Summary: Fluid Basics

Density =r=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+rgh
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
Bernoulli
p1+1/2rv12+rgy1=p2+1/2rv22+rgy2
Slow moving fluids exert more pressure than fast moving fluids