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Lecture 26

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Goals:

- Understand pressure in liquids
- Use Archimedes’ principle to understand buoyancy
- Understand the equation of continuity
- Use an ideal-fluid model to study fluid flow.

- At ordinary temperatures, matter exists in one of three states
- Solid: Has a shape and forms a surface
- Liquid: Has no shape but forms a surface
- Gas : No shape and does not form a surface

gas

liquid

solid

Increasing temperature

- Fluids: Liquids or Gases
- Shape determined by container
- Close contact at boundaries. No macroscopic gaps
- Forces exerted by fluids are always normal to the surface.
- Fluids can flow

- Fluids:
- They have mass
- They have weight
- They carry energy
- They have velocity when flow
- Newton’s Laws apply

- An intrinsic parameter of a fluid
- Density

units :

kg/m3 = 10-3 g/cm3

r(water) = 1.000 x 103 kg/m3 = 1.000 g/cm3

r(ice) = 0.917 x 103 kg/m3 = 0.917 g/cm3

r(air) = 1.29 kg/m3 = 1.29 x 10-3 g/cm3

r(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3

r(W or Au) = 19.3 x103 kg/m3 = 19.3 g/cm3

A

- Another parameter: Pressure
- A Fluid exerts force on the objects it contacts
- Molecules constantly bounces off the surface with F =Σ(Δmv/Δt) (momentum transfer / time)

- Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface.
- Force (avector) in a fluid can be expressed in terms of pressure (a scalar) as:

Do not confuse pressure with momentum

What is the SI unit of pressure?

- Pascal
- Atmosphere
- Bernoulli
- Young
- p.s.i.

Units :

1 N/m2 = 1 Pa (Pascal)

1 bar = 105 Pa

1 mbar = 102 Pa

1 torr = 133.3 Pa

1 atm = 1.013 x105 Pa

= 1013 mbar

= 760 Torr

= 14.7 lb/ in2 (=PSI)

- Distributing the force over a large area ….

- What is the initial force on the ping pong ball?
- Assuming this force is constant, how fast is the ping pong ball travelling when it leaves the tube?

F

This is a crude model….the speed of sound in air is 330 m/s

P0

F1

y1

y2

P1

A

P2

mg

F2

- When the pressure is small, relative to the bulk modulus of the fluid, we can treat the density as constant independent of pressure:
incompressible fluid

- For an incompressible fluid, the density is the same everywhere, but the pressure is NOT!
- P(y) = P0 - y g r
- Gauge pressure (subtract P0)
- PGauge = P(y) - P0

F2 = F1+ m g

= F1+ rVg

F2 /A = F1/A + rVg/A

P0

F1

y1

y2

P1

A

P2

mg

F2

Every 10 meters the pressure increases by 1 atm

- For a uniform fluid in an open container pressure same at a given depthindependentof the container

- Fluid level is the same everywhere in a connected container, assuming no surface forces

- Does PA=PB?

B

A

(A)r1 < r2

(B)r1 = r2

(C)r1>r2

- What happens with two fluids??
- Consider a U tube containing liquids of density r1 and r2 as shown:
Compare the densities of the liquids:

dI

r2

r1

- Invented by Torricelli
- A long closed tube is filled with mercury and inverted in a dish of mercury
- The closed end is nearly a vacuum
- PA=PB

- Measures atmospheric pressure as
1 atm = 0.760 m (of Hg)

- Consider the system shown:
- A downward force F1 is applied to the piston of area A1.
- This force is transmitted through the liquid to create an upward force F2.
- Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid.

P2

P1

P1 = P2

F1 / A1 = F2 / A2

A2 / A1 = F2 / F1

W2?

W1

W1 > W2

W1 < W2

W1 = W2

- Suppose we weigh an object in air (1) and in water (2).
How do these weights compare?

- Buoyant force = Weight of the fluid displaced

y

F

mg

B

- The buoyant force is equalto the weight of the liquid that is displaced.
- If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink.

- We can calculate how much of a floating object will be submerged in the liquid:

- Object is in equilibrium

piece of rock

on top of ice

What happens to the water level when the ice melts?

Expt. 1

Expt. 2

B. It stays the same

C. It drops

A. It rises

V1 = V2 = V3 = V4 = V5

m1 < m2 < m3 < m4 < m5

What is the final position of each block?

V1 = V2 = V3 = V4 = V5

m1 < m2 < m3 < m4 < m5

What is the final position of each block?

But this

Not this

- All of chapter 14