Pressure and Fluids

1
Pressure and Fluids
12.112.3
2
Density

• Relating how big to how much

12.1
3
Density

• Mass per volume r dm/dV m ?rdV
• If r is constant, m rV
• SI units kg/m3 conventionally g/cm3
• Intensive quantity
• Depends on substance, not amount

4
Densities
Substance kg/m3 g/cm3
Air 1.20 0.0012
Plain water 1,000 1.000
Sea water 1,030 1.03
Lead 11,300 11.3
5
Pressure

• One type of stress

12.2
6
Pressure

• Force per unit area p F?/A
• F? is component perpendicular to surface
• Scalar quantity

7
Units of Pressure

• SI Units N/m2 pascal Pa
• Bar 100,000 Pa mbar hPa 100 Pa
• Atmosphere atm mean sea level pressure
  101,325 Pa 1.01 bar
• Torr Height in mm of a mercury barometer (760
  torr 1 atm)
• lb/in2 psi (14.696 psi 1 atm)

8
Gauge and Absolute Pressure

• Gauge increase above surrounding atmosphere
• e.g. tire pressure 32 psig
• Absolute total

9
Poll Question

• When a molecule bounces elastically off a
  surface, what is the direction of its momentum
  change?

A. B.
C. D.
10
Poll Question

• When a molecule bounces elastically off a
  surface, what is the direction of the force on
  the surface?

A. B.
C. D.
11
Collisions

• Collision exerts a force into surface

• Pressure from force averaged over time and area

12
Pressure in a fluid

• Pascals Principle pressure in a fluid is
  exerted uniformly in all directions throughout
  the fluid.

13
Hydraulics

• just another simple machine

14
Quick Question

• If a force F1 is applied to the small piston with
  area A1, what is the pressure increase in the
  fluid?

1. Dp F1A1.
2. Dp F1 A1.
3. Dp A1/F1.
4. Dp F1/A1.

1
2
15
Poll Question

• If a force F1 is applied to the narrow piston
  with area A1, How does the magnitude of the force
  F2 exerted by the fluid on the wide piston
  compare to F1?

1. F2 F1.
2. F2 lt F1.
3. F2 gt F1.

1
2
16
Hydraulics

• Pistons have different areas
• Pressure p F/A for both pistons

• rearranges to

17
Poll Question

• If the narrow piston with area A1 is moved a
  distance Dx1, How does the distance Dx2 that the
  wide piston moves compare to Dx1?

1. Dx2 Dx1.
2. Dx2 lt Dx1.
3. Dx2 gt Dx1.

1
2
18
Hydraulics are Simple Machines

• Volume changes DV are opposites
• DV A1Dx1, so Dx1 DV/A1 Dx2 DV/A2

F1Dx1 F2Dx2

• input, output work equal and opposite

19
Example Problem

• In an auto shop a hydraulic jack is used to lift
  vehicles for service. If the radius of the
  cylinder below piston 1 is r1 0.02 m, and the
  radius of the cylinder below the piston holding
  the car is r2 0.1 m, what force F1 must be
  exerted on the small piston to lift a 1500-kg car?

20
Pressure with Depth
21
Pressure beneath a fluid

• Force comes from weight of fluid above
• Pressure increases with depth

22
Pressure in a liquid

• p F?/A

rgh
p pressure here
23
Depth Pressure Formula
p0 pressure here

• p p0 rgh

• p0 pressure at depth 0
• p pressure at depth h
• r constant density of liquid
• h depth under top of liquid

p pressure here
24
Buoyancy

• How do things float?

12.3
25
What forces are present?
26
What forces are present?
27
What forces are present?
28
What forces are present?
29
Pressure in a fluid

• Pressure increases with depth
• Greater pressure at bottom than top of an
  immersed object
• Results in upward buoyancy force that is the
  (vector) sum of all pA forces

30
Buoyancy from Ug

• Force dUg/dh
• Ug Ug obj Ug displaced fluid
• DUg mg(Dh) rVg(Dh)
• ?F dUg/dh mg rVg
• mg gravity force
• rVg buoyancy force

31
Buoyancy Force

• Buoyancy force weight of fluid displaced
• F rVg
• r constant density of fluid
• V volume of fluid displaced volume of object
  submerged
• g 9.8 N/kg

32
Quick example

• The density of fresh water r 1000 kg/m3.
• What is the buoyancy force on a 1-m3 parcel of
  water?
• What is the buoyancy force on a 1-m3 rock
  submerged under water?
• What is the buoyancy force on a 1-m3 chunk of
  Styrofoam submerged under water?

33
Quick Questions

• All blocks are cubes with 1-m sides.

Which ones float? Which one makes the water level
change the most? Which one makes the water level
change the least?
34
Buoyancy and Density

• If an object is more dense than the surrounding
  fluid, it sinks
• If an object is less dense than the surrounding
  fluid, it rises
• A floating object displaces exactly enough fluid
  to support its weight

35
Quick Question

• Which boat (with cargo) has the greatest volume?
• The high boat (left).
• The low boat (right).
• Both have the same volume.

36
Quick Question

• Which boat (with cargo) has the greatest volume
  under the water line?
• The high boat (left).
• The low boat (right).
• Both have the same volume under water.

37
Quick Question

• Which boat (with cargo) experiences the greatest
  buoyancy force?
• The high boat (left).
• The low boat (right).
• Both experience the same buoyancy force.

38
Quick Question

• Which boat (with cargo) has the greatest mass?
• The high boat (left).
• The low boat (right).
• Both have the same mass.

39
Poll Question

• After a boat sinks, it displaces
• more water than when it floated.
• less water than when it floated.
• the same volume of water as when it floated.