Pressure and Fluids

§ 12.1–12.3

Density

Relating “how big” to “how much”

§ 12.1

Density

• Mass per volume = dm/dV; m = dV

• If is constant, m = V

• SI units kg/m3; conventionally g/cm3

• Intensive quantity– Depends on substance, not amount

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

Pressure

One type of stress

§ 12.2

Pressure

• Force per unit area p = F/A

• F is component perpendicular to surface

• Scalar quantity

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)

Gauge and Absolute Pressure

• Gauge: increase above surrounding atmosphere– e.g. tire pressure 32 psig

• Absolute: total

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

A.

B.

C.

D.

Poll QuestionWhen a molecule bounces elastically off a surface, what is the direction of the force on the surface?

A.

B.

C.

D.

Collisions

• Collision exerts a force into surface

• Pressure from force averaged over time and area

Force

Pressure in a fluid

• Pascal’s Principle: pressure in a fluid is exerted uniformly in all directions throughout the fluid.

Hydraulics

just another simple machine

Quick Question

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

A. p = F1A1.

B. p = F1 + A1.

C. p = A1/F1.

D. p = F1/A1.1 2

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?

A. F2 = F1.

B. F2 < F1.

C. F2 > F1.

1 2

Hydraulics

• Pistons have different areas

• Pressure p = F/A for both pistons

F1

A1

F2

A2=

F1

F2

A1

A2=

• rearranges to

Poll Question

If the narrow piston with area A1 is moved a distance x1, How does the distance x2 that the wide piston moves compare to x1?

A. x2 = x1.

B. x2 < x1.

C. x2 > x1.

1 2

x1

x2 ?

Hydraulics are Simple Machines

• Volume changes V are opposites

• V = A1x1, so x1 = V/A1; x2 = –V/A2

F1

A1

F2

A2=

F1x1 = –F2x2

• input, output work equal and opposite

=VA1

F1

VA2

F2

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?

Pressure with Depth

Pressure beneath a fluid

• Force comes from weight of fluid above

• Pressure increases with depth

Supports weight above

Pressure in a liquid

p = F/A =

gh

weight of fluid abovecross-section

=mass g

Adensity volume g

A=

AAhg

=

h

p = pressure here

Depth Pressure Formula

p = p0 + gh

• p0 = pressure at depth 0

• p = pressure at depth h

• = constant density of liquid

• h = depth under top of liquid

h

p = pressure here

p0 = pressure here

Buoyancy

How do things float?

§ 12.3

What forces are present?

What forces are present?

What forces are present?

What forces are present?

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

F = p dAA

Buoyancy from Ug

Force = –dUg/dh

Ug = Ug obj + Ug displaced fluid

Ug = mg(h) + Vg(–h)

∑F = –dUg/dh = –mg + Vg

–mg = gravity force

Vg = buoyancy force

h

Quick example

The density of fresh water = 1000 kg/m3.a. What is the buoyancy force on a 1-m3 parcel

of water?

b. What is the buoyancy force on a 1-m3 rock submerged under water?

c. What is the buoyancy force on a 1-m3 chunk of Styrofoam submerged under water?

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?

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

Quick Question

Which boat (with cargo) has the greatest volume?

A. The high boat (left).

B. The low boat (right).

C. Both have the same volume.

Quick QuestionWhich boat (with cargo) has the greatest volume under the

water line?

A. The high boat (left).

B. The low boat (right).

C. Both have the same volume under water.

Quick QuestionWhich boat (with cargo) experiences the greatest buoyancy

force?

A. The high boat (left).

B. The low boat (right).

C. Both experience the same buoyancy force.

Quick Question

Which boat (with cargo) has the greatest mass?

A. The high boat (left).

B. The low boat (right).

C. Both have the same mass.

Poll Question

After a boat sinks, it displaces

A. more water than when it floated.

B. less water than when it floated.

C. the same volume of water as when it floated.