Archimedes’ Principle

Physics 202Professor Lee

CarknerLecture 2

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

pressure P = gh

P = = 1000 kg/m3

g = 9.8 m/s2

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

On Mars pressure would decrease

Mars has smaller value of g

Archimedes’ 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,

iVig=wVwg

Vw=Vi (i/w)

w = 1024 kg/m3 and i = 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?

Av = 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 Can’t 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

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

at the far end with fluid being forced through it

Wg = -mg(y2-y1) = -gV(y2-y1) The work of the system due to pressure is,

Wp=Fd=pAd=pV=-(p2-p1)V

(1/2mv2)=1/2V(v22-v1

2)

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

2+gy2

Consequences of Bernoulli’s

Fast moving fluids exert less pressure than slow moving fluids

This is known as Bernoulli’s 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 Bernoulli’s equation:

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

2

The difference in pressure is:pb-pt=1/2vt

2-1/2vb2

(Fb/A)-(Ft/A)=1/2(vt

2-vb2)

L= (½)A(vt

2-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?a) Doubling your massb) Doubling the mass of the earthc) Doubling your heightd) Doubling the size of your

shoese) Doubling air pressure

Which of the following would increase the pressure of a column of fluid of fixed mass the most?a) Doubling the width of the columnb) Halving the density of the fluidc) Halving the mass of the Earthd) Halving the speed of the Earth’s

rotatione) Doubling the height of the

column

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