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

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,

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

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 = -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 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+1/2rvt2=pb+1/2rvb2

The difference in pressure is:

pb-pt=1/2rvt2-1/2rvb2

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

L= (½)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 Earth’s 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

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