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Cube A has edge length L and mass M. Cube B has edge length 2L and mass 4M. Which has greater density?

Clicker Question Room Frequency BA

Cube A has larger density. ρA = M/L3 , ρB = (4M)/(2L)3 = (1/2)M/L3 so object A has twice the density of object B.

A) A has larger densityB) B has larger densityC) A and B have the same density.

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• CAPA assignment #12 is due on Friday at 10 pm.• Next week in Section: Assignment #6• Finish reading Chapter 10 on Fluids• Midterm 3 scores are uploaded on CU Learn. Solutions

are posted on CULearn. Rough grade scale is

Announcements

Average = 62.2 out of 100Stan. Dev. = 17.3A range (80-100) B range (60-80)C range (45-60) D range (30-45)F range (0-30)

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Fluid DensityWe imagine that the volume V is completely filled with some “continuous” substance. We measure the mass m and the volume V then calculate the mass density ρmass as m/V.

Gases: Compressible (expand or compress)Liquids: Often nearly incompressible

We’ll typically assume, for a given substance, that the mass density is the same everywhere in side the fluid! This is a good assumption for statics.

SI Units are kg/m3 =1000 g/cm3

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Specific GravityAnother common and useful measure of density is the specific gravity (SG).

SGWater = 1SGIron = 7.9SGPb = 11.9SGIce = 0.92SGAlcohol= 0.79

No “visible” units because it is a ratio!

Specific Gravity (SG) of substance X ≡ρX

ρWateρ

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Back to square 2: Fluid “Forces”

When we try to apply Newton’s Laws to Fluids what do we use for the force?

Again we consider a “small” imaginary volume or box inside the body of fluid.

To find the force on the imaginary volume V, we use the concept of pressure. We find the force F on each side of the volume V and divide the magnitude of that force by the area A of that side. The pressure P is then F/A

Pat side of V =

ρFon side of V

Aof side V

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Units of Pressure

Pat side of V =

ρFon side of V

Aof side V

SI Units of Pressure: N/m2 = 1 Pascal (Pa)

English Units of Pressure: lb/in2 = 1 pound per square inch (psi)

How do these unit compare in size?

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Approximately(!) what is 1 psi in the SI unit Pa? (1 lb = 4.45 N)

Clicker Question Room Frequency BA

A) 5B) 200C) 7000D) 4 x 104

E) 1 x 105

Both force and area units have to be converted!

Guess estimate 1 lb

in2 ≈5 Nlb

g40 in1 m

⎛⎝⎜

⎞⎠⎟2

=8000Nm 2

Accurate Calculation 1

lbin2 =

4.45 Nlb

g39.4 in1 m

⎛⎝⎜

⎞⎠⎟2

=6910Nm 2

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Atomic View of Fluid Pressure from Air

Air consists mostly of oxygen and nitrogen molecules. At room temperature, the molecules have thermal energy and are moving around rapidly (speed ≈ 400 m/s), colliding with each other and with every exposed surface. The pounding of the air molecules on a surface, like the pitter-pat of rain on the roof, adds up to a large force per area: Patm = 14.7 psi.

In a later chapter we’ll calculate the force on the walls directly from these atomic collisions!

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Too Many Units of Pressure!

The most common pressure in everyday life is that coming from the atmosphere (at sea level) !

1 atmosphere = 1 atm = 14.7 psi = 1.013 x 105 Pa

= 1.013 bar = 760 torr

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The air pressure inside the Space Station is P = 12 psi. There are two square windows in the Space Station: a little one and a big one. The big window is 30 cm on a side. The little window is 15 cm on a side. How does the pressure on the big window compare to the pressure on the little window?

Clicker Question Room Frequency BA

A) same pressure on both windows B) 2 times more pressure on the big window C) 4 times more pressure on the big window D) 9 times more pressure on the big window

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The air pressure inside the Space Station is P = 12 psi. There are two square windows in the Space Station: a little one and a big one. The big window is 30 cm on a side. The little window is 15 cm on a side. How does the total force on the big window compare to the total force on the little window?

A) same force on both windows B) 2 times more force on the big window C) 4 times more force on the big window D) 9 times more force on the big window

Clicker Question Room Frequency BA

F = P*A

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Direction of Fluid Forces

Newton’s law deals with forces which are vectors! What is the direction of the force from fluid pressure?

For static fluids (no flow) the force is perpendicular to the surface of the side.

In the figure you see arrows for the force on each side of an imaginary cube of volume V.

Critical point: Pressure does not have any direction; the direction of the force from pressure depends on the orientation of the surface the pressure acts on.

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Is 15 psi a big pressure?

Yes!!!

We are normally unaware of this because forces on us are balanced.

Here are two video examples and a demo!

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Pascal’s Principle: If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount.

POUT =PIN

FOUTAOUT

=FIN

AIN

FOUT =AOUT

AIN

⎛⎝⎜

⎞⎠⎟FIN

Example: Hydraulic lift

Applying Outside Pressure to a Fluid

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Pressure in Fluids on a Planet I

For our first application of Newton’s 2nd Law on Fluids, let’s consider the result of gravity on the pressure in a fluid.

Consider an imaginary cube with each side of area A, the top at the surface and the bottom at depth h.

Each of the six faces contributes to the net force!

Four side faces cancel each other; what about top and bottom faces?

Fnet = Ftop + Fbottom = 0 Ph

Patm

Fnet = -PatmA + PhA - Mwaterg = 0

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Pressure in Fluids on a Planet II

Ph

Patm

Fnet = -PatmA + PhA - Mwaterg = 0

PhA =PatmA+MWateρg

Ph =Patm +MWateρg

A

Ph =Patm +ρWateρ(Ah)g

A =Patm + ρWateρhg

Pressure increases linearly with depth, proportional to g and ρ!

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More on Gravity and Pascal’s Principle

Pressure only depends on the depth; at a given depth the pressure is the same.

Gauge Pressure is the amount of pressure after subtracting Patm

Often people leave out the Patm, but if you want the total, absolute pressure, you must include it.

Ph =Patm + ρWateρhg

How do you know what kind of gauge you are reading?

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Example. At the surface of a swimming pool the pressure on a swimmer is due to air: Patm = 1 atm. At what depth in the pool will the pressure be 2 atm?

Ph =Patm + ρgh

1 atm ~ 105 Pa

h =Ph −Patm

ρg

h =2x105 −1x105 Pa

(1000kg/m 3)(10 m /s2≈10 m

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P =ρgh

As shown, two containers are connected by a hose and are filled with water. Which picture correctly depicts the water levels?

Clicker Question Room Frequency BA

Different depths would give different pressures! The net force on the fluid at the connection tube would not be zero and there would be flow!