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Archimedes 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

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,
- riVigrwVwg
- VwVi (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

- RAvconstant is called the equation of

continuity - You can use it to determine the flow rates of a

system of pipes - Cant 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

Bernoullis 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,
- WpFdpAdDpDV-(p2-p1)DV
- D(1/2mv2)1/2rDV(v22-v12)
- p1(1/2)rv12rgy1p2(1/2)rv22rgy2

Consequences of Bernoullis

- Fast moving fluids exert less pressure than slow

moving fluids - This is known as Bernoullis 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 Bernoullis equation
- pt1/2rvt2pb1/2rvb2
- The difference in pressure is
- pb-pt1/2rvt2-1/2rvb2
- (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 Earths rotation
- Doubling the height of the column

Summary Fluid Basics

- Density rm/V
- PressurepF/A
- On Earth the atmosphere exerts a pressure and

gravity causes columns of fluid to exert pressure - Pressure of column of fluid
- pp0rgh
- 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 - FoFi(Ao/Ai) and dodi(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 (RAv) is a

constant - fluid moving into a narrower pipe speeds up
- Bernoulli
- p11/2rv12rgy1p21/2rv22rgy2
- Slow moving fluids exert more pressure than fast

moving fluids